Distance on a great circle

Percentage Accurate: 61.8% → 98.6%

Time: 34.2s

Alternatives: 34

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 60.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 61.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 76.5%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

11. Applied rewrites 77.6%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

13. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-neg.f64 98.6%

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

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

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

    16. metadata-eval N/A

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

    17. lower-*.f64 98.6%

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

    18. lift-*.f64 N/A

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

15. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-neg.f64 98.6%

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

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

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

    16. metadata-eval N/A

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

    17. lower-*.f64 98.6%

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

    18. lift-*.f64 N/A

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

17. Applied rewrites 98.6%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 60.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 61.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 76.5%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

11. Applied rewrites 77.6%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

13. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-neg.f64 98.6%

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

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

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

    16. metadata-eval N/A

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

    17. lower-*.f64 98.6%

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

    18. lift-*.f64 N/A

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

15. Applied rewrites 98.6%

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

    1. lift--.f64 N/A

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

    2. lift-*.f64 N/A

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

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

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. lift-*.f64 N/A

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

    8. *-commutative N/A

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

    9. lower-*.f64 N/A

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

    10. lower-neg.f64 98.6%

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

    11. lift-cos.f64 N/A

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

    12. cos-neg-rev N/A

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

    13. lower-cos.f64 N/A

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

    14. lift-*.f64 N/A

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

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

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

    16. metadata-eval N/A

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

    17. lower-*.f64 98.6%

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

    18. lift-*.f64 N/A

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

17. Applied rewrites 98.6%

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

18. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

20. Applied rewrites 98.6%

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

21. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

23. Applied rewrites 98.6%

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

Alternative 3: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 60.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 61.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 76.5%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

11. Applied rewrites 77.6%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

13. Applied rewrites 98.6%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

16. Applied rewrites 98.6%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-*.f64 N/A

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

    2. lower-cos.f64 N/A

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

    3. lower-*.f64 N/A

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

    4. lower-cos.f64 N/A

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

    5. lower-pow.f64 N/A

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

19. Applied rewrites 98.6%

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

Alternative 4: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

3. Applied rewrites 60.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

5. Applied rewrites 62.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

7. Applied rewrites 61.9%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. sub-flip N/A

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

   5. div-add N/A

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

   6. sin-sum N/A

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

   7. lower-fma.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. mult-flip N/A

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

   10. metadata-eval N/A

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

   11. lower-*.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. frac-2neg-rev N/A

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

   16. lower-/.f64 N/A

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

   17. metadata-eval N/A

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

   18. lower-*.f64 N/A

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

   19. lower-cos.f64 N/A

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

   20. mult-flip N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. lower-sin.f64 N/A

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

   24. metadata-eval N/A

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

   25. metadata-eval N/A

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

9. Applied rewrites 76.5%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

11. Applied rewrites 77.6%

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

    1. lift-sin.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift--.f64 N/A

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

    4. div-sub N/A

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

    5. sin-diff N/A

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

    6. lower--.f64 N/A

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

13. Applied rewrites 98.6%

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

14. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

16. Applied rewrites 98.6%

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

17. Taylor expanded in lambda1 around inf

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

    1. lower-fma.f64 N/A

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

19. Applied rewrites 98.6%

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

Alternative 5: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ t_2 := \sqrt{1 - t\_1}\\ t_3 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\\ \mathbf{if}\;\lambda_2 \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{t\_2}\right)\\ \mathbf{elif}\;\lambda_2 \leq 15000000000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{{\left({t\_1}^{0.25}\right)}^{2}}{t\_2}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* (- lambda1 lambda2) 0.5)))
        (t_1
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
        (t_2 (sqrt (- 1.0 t_1)))
        (t_3
         (+
          (pow
           (-
            (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
            (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
           2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))
   (if (<= lambda2 -2.7e-5)
     (* R (* 2.0 (atan2 (sqrt t_1) t_2)))
     (if (<= lambda2 15000000000.0)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (pow (pow t_1 0.25) 2.0) t_2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) * 0.5));
	double t_1 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0)), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	double t_2 = sqrt((1.0 - t_1));
	double t_3 = pow(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0);
	double tmp;
	if (lambda2 <= -2.7e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_1), t_2));
	} else if (lambda2 <= 15000000000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(pow(pow(t_1, 0.25), 2.0), t_2));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) * 0.5))
	t_1 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	t_2 = sqrt(Float64(1.0 - t_1))
	t_3 = Float64((Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))
	tmp = 0.0
	if (lambda2 <= -2.7e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), t_2)));
	elseif (lambda2 <= 15000000000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(((t_1 ^ 0.25) ^ 2.0), t_2)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, -2.7e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 15000000000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Power[N[Power[t$95$1, 0.25], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -2.6999999999999999e-5

   1. Initial program 61.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 60.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 62.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   7. Applied rewrites 61.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   9. Applied rewrites 76.5%

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

   10. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   12. Applied rewrites 76.5%

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

   13. Taylor expanded in lambda1 around inf

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

       1. lower-fma.f64 N/A

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

   15. Applied rewrites 76.5%

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

   if -2.6999999999999999e-5 < lambda2 < 1.5e10

   1. Initial program 61.8%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   3. Applied rewrites 60.9%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. sub-flip N/A

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

      5. div-add N/A

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

      6. sin-sum N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. mult-flip N/A

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. metadata-eval N/A

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

      14. metadata-eval N/A

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

      15. frac-2neg-rev N/A

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

      16. lower-/.f64 N/A

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

      17. metadata-eval N/A

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

      18. lower-*.f64 N/A

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

      19. lower-cos.f64 N/A

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

      20. mult-flip N/A

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

      21. metadata-eval N/A

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

      22. lower-*.f64 N/A

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

      23. lower-sin.f64 N/A

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

      24. metadata-eval N/A

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

      25. metadata-eval N/A

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

   5. Applied rewrites 62.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 77.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right) + \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right)} \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right)} \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. sin-sum-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{-2} + \lambda_1 \cdot \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. +-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \frac{\lambda_2}{-2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2} + \color{blue}{\frac{\lambda_2}{-2}}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\lambda_1 \cdot \frac{1}{2}} + \frac{\lambda_2}{-2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}} + \frac{\lambda_2}{-2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{\lambda_1}{2}} + \frac{\lambda_2}{-2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. frac-2neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(-2\right)}}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{2}}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. div-add N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. sub-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 78.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right) + \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right)} \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right)} \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. sin-sum-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{-2} + \lambda_1 \cdot \frac{1}{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. +-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \frac{\lambda_2}{-2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2} + \color{blue}{\frac{\lambda_2}{-2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\color{blue}{\lambda_1 \cdot \frac{1}{2}} + \frac{\lambda_2}{-2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}} + \frac{\lambda_2}{-2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{\lambda_1}{2}} + \frac{\lambda_2}{-2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. frac-2neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(-2\right)}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. div-add N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}{2}\right)}}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. sub-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 79.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   18. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right) + \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right)} \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right)} \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. sin-sum-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{-2} + \lambda_1 \cdot \frac{1}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. +-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \frac{\lambda_2}{-2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2} + \color{blue}{\frac{\lambda_2}{-2}}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\lambda_1 \cdot \frac{1}{2}} + \frac{\lambda_2}{-2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}} + \frac{\lambda_2}{-2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\color{blue}{\frac{\lambda_1}{2}} + \frac{\lambda_2}{-2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. frac-2neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(-2\right)}}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{2}}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. div-add N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}{2}\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       19. sub-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       20. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   19. Applied rewrites 78.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   20. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right) + \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

       2. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \sin \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       4. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       6. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\color{blue}{\sin \left(\frac{\lambda_2}{-2}\right)} \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\cos \left(\lambda_1 \cdot \frac{1}{2}\right)} + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       8. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \color{blue}{\cos \left(\frac{\lambda_2}{-2}\right)} \cdot \sin \left(\lambda_1 \cdot \frac{1}{2}\right)\right)\right)}}\right) \]

       9. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \left(\sin \left(\frac{\lambda_2}{-2}\right) \cdot \cos \left(\lambda_1 \cdot \frac{1}{2}\right) + \cos \left(\frac{\lambda_2}{-2}\right) \cdot \color{blue}{\sin \left(\lambda_1 \cdot \frac{1}{2}\right)}\right)\right)}}\right) \]

       10. sin-sum-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_2}{-2} + \lambda_1 \cdot \frac{1}{2}\right)}\right)}}\right) \]

       11. +-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2} + \frac{\lambda_2}{-2}\right)}\right)}}\right) \]

       12. lift-/.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\lambda_1 \cdot \frac{1}{2} + \color{blue}{\frac{\lambda_2}{-2}}\right)\right)}}\right) \]

       13. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\color{blue}{\lambda_1 \cdot \frac{1}{2}} + \frac{\lambda_2}{-2}\right)\right)}}\right) \]

       14. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}} + \frac{\lambda_2}{-2}\right)\right)}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\color{blue}{\frac{\lambda_1}{2}} + \frac{\lambda_2}{-2}\right)\right)}}\right) \]

       16. frac-2neg N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \color{blue}{\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(-2\right)}}\right)\right)}}\right) \]

       17. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{2}}\right)\right)}}\right) \]

       18. div-add N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}{2}\right)}\right)}}\right) \]

       19. sub-flip N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

       20. lift--.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   21. Applied rewrites 78.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right) \cdot \color{blue}{\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}\right)}}\right) \]

   if 1.5e10 < lambda2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   17. Applied rewrites 41.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(\cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \color{blue}{\tan^{-1}_* \frac{{\left({\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{2}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   20. Applied rewrites 76.4%

       \[\leadsto R \cdot \left(2 \cdot \color{blue}{\tan^{-1}_* \frac{{\left({\left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_0\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ t_2 := \sqrt{1 - t\_1}\\ t_3 := {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {t\_0}^{2}\right)\\ \mathbf{if}\;\lambda_1 \leq -0.0042:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{{\left({t\_1}^{0.25}\right)}^{2}}{t\_2}\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.3 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{t\_2}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 lambda2)))
        (t_1
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) t_0))
            2.0))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
        (t_2 (sqrt (- 1.0 t_1)))
        (t_3
         (+
          (pow
           (fma
            (sin (* phi2 0.5))
            (- (cos (* -0.5 phi1)))
            (* (cos (* -0.5 phi2)) (sin (* phi1 0.5))))
           2.0)
          (* (cos phi1) (* (cos phi2) (pow t_0 2.0))))))
   (if (<= lambda1 -0.0042)
     (* R (* 2.0 (atan2 (pow (pow t_1 0.25) 2.0) t_2)))
     (if (<= lambda1 3.3e-13)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       (* R (* 2.0 (atan2 (sqrt t_1) t_2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * lambda2));
	double t_1 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * t_0)), 2.0)), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	double t_2 = sqrt((1.0 - t_1));
	double t_3 = pow(fma(sin((phi2 * 0.5)), -cos((-0.5 * phi1)), (cos((-0.5 * phi2)) * sin((phi1 * 0.5)))), 2.0) + (cos(phi1) * (cos(phi2) * pow(t_0, 2.0)));
	double tmp;
	if (lambda1 <= -0.0042) {
		tmp = R * (2.0 * atan2(pow(pow(t_1, 0.25), 2.0), t_2));
	} else if (lambda1 <= 3.3e-13) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), t_2));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * lambda2))
	t_1 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * t_0)) ^ 2.0)), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	t_2 = sqrt(Float64(1.0 - t_1))
	t_3 = Float64((fma(sin(Float64(phi2 * 0.5)), Float64(-cos(Float64(-0.5 * phi1))), Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(phi1 * 0.5)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (t_0 ^ 2.0))))
	tmp = 0.0
	if (lambda1 <= -0.0042)
		tmp = Float64(R * Float64(2.0 * atan(((t_1 ^ 0.25) ^ 2.0), t_2)));
	elseif (lambda1 <= 3.3e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), t_2)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.0042], N[(R * N[(2.0 * N[ArcTan[N[Power[N[Power[t$95$1, 0.25], $MachinePrecision], 2.0], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 3.3e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -0.0041999999999999997

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   17. Applied rewrites 41.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{{\left(\mathsf{fma}\left(\cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right)}^{\left(\frac{0.5}{2}\right)} \cdot {\left(\mathsf{fma}\left(\cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot \lambda_2\right)\right)\right), \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)\right)\right)\right)}^{\left(\frac{0.5}{2}\right)}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \color{blue}{\tan^{-1}_* \frac{{\left({\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}^{\frac{1}{4}}\right)}^{2}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   20. Applied rewrites 76.4%

       \[\leadsto R \cdot \left(2 \cdot \color{blue}{\tan^{-1}_* \frac{{\left({\left(\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\right)}^{0.25}\right)}^{2}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   if -0.0041999999999999997 < lambda1 < 3.3000000000000001e-13

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 77.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   18. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   19. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 57.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   20. Applied rewrites 57.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   21. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]
   22. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

       7. lower-*.f64 57.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

   23. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]

   if 3.3000000000000001e-13 < lambda1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 76.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 76.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(-0.5 \cdot \lambda_2\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot t\_0\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ t_3 := {\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {t\_0}^{2}\right)\\ \mathbf{if}\;\lambda_1 \leq -0.0042:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\lambda_1 \leq 3.3 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* -0.5 lambda2)))
        (t_1
         (fma
          (cos phi1)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) t_0))
            2.0))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
        (t_2 (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))))))
        (t_3
         (+
          (pow
           (fma
            (sin (* phi2 0.5))
            (- (cos (* -0.5 phi1)))
            (* (cos (* -0.5 phi2)) (sin (* phi1 0.5))))
           2.0)
          (* (cos phi1) (* (cos phi2) (pow t_0 2.0))))))
   (if (<= lambda1 -0.0042)
     t_2
     (if (<= lambda1 3.3e-13)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((-0.5 * lambda2));
	double t_1 = fma(cos(phi1), (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * t_0)), 2.0)), pow(sin((0.5 * (phi1 - phi2))), 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	double t_3 = pow(fma(sin((phi2 * 0.5)), -cos((-0.5 * phi1)), (cos((-0.5 * phi2)) * sin((phi1 * 0.5)))), 2.0) + (cos(phi1) * (cos(phi2) * pow(t_0, 2.0)));
	double tmp;
	if (lambda1 <= -0.0042) {
		tmp = t_2;
	} else if (lambda1 <= 3.3e-13) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * lambda2))
	t_1 = fma(cos(phi1), Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * t_0)) ^ 2.0)), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))))
	t_3 = Float64((fma(sin(Float64(phi2 * 0.5)), Float64(-cos(Float64(-0.5 * phi1))), Float64(cos(Float64(-0.5 * phi2)) * sin(Float64(phi1 * 0.5)))) ^ 2.0) + Float64(cos(phi1) * Float64(cos(phi2) * (t_0 ^ 2.0))))
	tmp = 0.0
	if (lambda1 <= -0.0042)
		tmp = t_2;
	elseif (lambda1 <= 3.3e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision]) + N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -0.0042], t$95$2, If[LessEqual[lambda1, 3.3e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -0.0041999999999999997 or 3.3000000000000001e-13 < lambda1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 76.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around inf

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 76.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   if -0.0041999999999999997 < lambda1 < 3.3000000000000001e-13

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 77.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   18. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   19. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 57.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   20. Applied rewrites 57.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   21. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]
   22. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

       7. lower-*.f64 57.0%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)\right)}}\right) \]

   23. Applied rewrites 57.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}\right)}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 85.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_2 := {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\\ t_3 := \mathsf{fma}\left(t\_0 \cdot \cos \phi_2, \cos \phi_1, t\_2\right)\\ t_4 := \mathsf{fma}\left(t\_0, \cos \phi_2 \cdot \cos \phi_1, t\_2\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\\ \mathbf{elif}\;\phi_1 \leq 0.049:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5)))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_2
         (pow
          (fma
           (- (cos (* phi1 -0.5)))
           (sin (* 0.5 phi2))
           (* (sin (* phi1 0.5)) (cos (* -0.5 phi2))))
          2.0))
        (t_3 (fma (* t_0 (cos phi2)) (cos phi1) t_2))
        (t_4 (fma t_0 (* (cos phi2) (cos phi1)) t_2)))
   (if (<= phi1 -2e-7)
     (* (* R 2.0) (atan2 (sqrt t_4) (sqrt (- 1.0 t_4))))
     (if (<= phi1 0.049)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_2 = pow(fma(-cos((phi1 * -0.5)), sin((0.5 * phi2)), (sin((phi1 * 0.5)) * cos((-0.5 * phi2)))), 2.0);
	double t_3 = fma((t_0 * cos(phi2)), cos(phi1), t_2);
	double t_4 = fma(t_0, (cos(phi2) * cos(phi1)), t_2);
	double tmp;
	if (phi1 <= -2e-7) {
		tmp = (R * 2.0) * atan2(sqrt(t_4), sqrt((1.0 - t_4)));
	} else if (phi1 <= 0.049) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_2 = fma(Float64(-cos(Float64(phi1 * -0.5))), sin(Float64(0.5 * phi2)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(-0.5 * phi2)))) ^ 2.0
	t_3 = fma(Float64(t_0 * cos(phi2)), cos(phi1), t_2)
	t_4 = fma(t_0, Float64(cos(phi2) * cos(phi1)), t_2)
	tmp = 0.0
	if (phi1 <= -2e-7)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4))));
	elseif (phi1 <= 0.049)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[((-N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision]) * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]}, If[LessEqual[phi1, -2e-7], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 0.049], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.9999999999999999e-7

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 77.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   19. Applied rewrites 75.9%

       \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}} \]

   if -1.9999999999999999e-7 < phi1 < 0.049000000000000002

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 62.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 60.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]

   if 0.049000000000000002 < phi1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 77.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   19. Applied rewrites 76.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   21. Applied rewrites 75.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5\right) \cdot \cos \phi_2, \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)\\ t_2 := \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_1 \leq 0.049:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi2)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_1
         (fma
          (- 0.5 (* (cos (* 1.0 (- lambda1 lambda2))) 0.5))
          (* (cos phi2) (cos phi1))
          (pow
           (fma
            (- (cos (* phi1 -0.5)))
            (sin (* 0.5 phi2))
            (* (sin (* phi1 0.5)) (cos (* -0.5 phi2))))
           2.0)))
        (t_2 (* (* R 2.0) (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))))))
   (if (<= phi1 -2e-7)
     t_2
     (if (<= phi1 0.049)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi2) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_1 = fma((0.5 - (cos((1.0 * (lambda1 - lambda2))) * 0.5)), (cos(phi2) * cos(phi1)), pow(fma(-cos((phi1 * -0.5)), sin((0.5 * phi2)), (sin((phi1 * 0.5)) * cos((-0.5 * phi2)))), 2.0));
	double t_2 = (R * 2.0) * atan2(sqrt(t_1), sqrt((1.0 - t_1)));
	double tmp;
	if (phi1 <= -2e-7) {
		tmp = t_2;
	} else if (phi1 <= 0.049) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi2) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_1 = fma(Float64(0.5 - Float64(cos(Float64(1.0 * Float64(lambda1 - lambda2))) * 0.5)), Float64(cos(phi2) * cos(phi1)), (fma(Float64(-cos(Float64(phi1 * -0.5))), sin(Float64(0.5 * phi2)), Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(-0.5 * phi2)))) ^ 2.0))
	t_2 = Float64(Float64(R * 2.0) * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))))
	tmp = 0.0
	if (phi1 <= -2e-7)
		tmp = t_2;
	elseif (phi1 <= 0.049)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[Power[N[((-N[Cos[N[(phi1 * -0.5), $MachinePrecision]], $MachinePrecision]) * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2e-7], t$95$2, If[LessEqual[phi1, 0.049], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.9999999999999999e-7 or 0.049000000000000002 < phi1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   11. Applied rewrites 77.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   12. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1 - \phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\color{blue}{\phi_1 - \phi_2}}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \color{blue}{\left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   15. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   16. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) + \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right) + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\color{blue}{\sin \left(\frac{1}{2} \cdot \phi_2\right) \cdot \left(\mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)} + \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\frac{1}{2} \cdot \phi_2\right), \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\frac{1}{2} \cdot \phi_2\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}, \mathsf{neg}\left(\cos \left(\frac{1}{2} \cdot \phi_1\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-neg.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), \color{blue}{-\cos \left(0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lift-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\frac{1}{2} \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       12. cos-neg-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       13. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\color{blue}{\cos \left(\mathsf{neg}\left(\frac{1}{2} \cdot \phi_1\right)\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       14. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\mathsf{neg}\left(\color{blue}{\frac{1}{2} \cdot \phi_1}\right)\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       15. distribute-lft-neg-in N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \color{blue}{\left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \phi_1\right)}, \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\color{blue}{\frac{-1}{2}} \cdot \phi_1\right), \sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       17. lower-*.f64 98.6%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \color{blue}{\left(-0.5 \cdot \phi_1\right)}, \sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       18. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \cos \left(\frac{-1}{2} \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot \frac{1}{2}\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot \frac{1}{2}\right), -\cos \left(\frac{-1}{2} \cdot \phi_1\right), \color{blue}{\sin \left(\frac{1}{2} \cdot \phi_1\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right)}\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   17. Applied rewrites 98.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\color{blue}{\left(\mathsf{fma}\left(\sin \left(\phi_2 \cdot 0.5\right), -\cos \left(-0.5 \cdot \phi_1\right), \cos \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)\right)}}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   19. Applied rewrites 75.9%

       \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - \cos \left(1 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot 0.5, \cos \phi_2 \cdot \cos \phi_1, {\left(\mathsf{fma}\left(-\cos \left(\phi_1 \cdot -0.5\right), \sin \left(0.5 \cdot \phi_2\right), \sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(-0.5 \cdot \phi_2\right)\right)\right)}^{2}\right)}}} \]

   if -1.9999999999999999e-7 < phi1 < 0.049000000000000002

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 62.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 60.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_2 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 77.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_2 := \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{t\_0}\right) \cdot t\_0\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 3400000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (- phi2)))))
        (t_1
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (*
           (cos phi1)
           (pow
            (fma
             (cos (* -0.5 lambda2))
             (sin (* 0.5 lambda1))
             (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
            2.0))))
        (t_2
         (*
          (+
           1.0
           (/
            (*
             (cos phi2)
             (-
              0.5
              (*
               0.5
               (fma
                (cos lambda1)
                (cos lambda2)
                (* (sin lambda1) (sin lambda2))))))
            t_0))
          t_0))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -0.37)
     t_3
     (if (<= phi2 3400000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos(-phi2));
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0));
	double t_2 = (1.0 + ((cos(phi2) * (0.5 - (0.5 * fma(cos(lambda1), cos(lambda2), (sin(lambda1) * sin(lambda2)))))) / t_0)) * t_0;
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = t_3;
	} else if (phi2 <= 3400000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(-phi2))))
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0)))
	t_2 = Float64(Float64(1.0 + Float64(Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * fma(cos(lambda1), cos(lambda2), Float64(sin(lambda1) * sin(lambda2)))))) / t_0)) * t_0)
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = t_3;
	elseif (phi2 <= 3400000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[(-phi2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], t$95$3, If[LessEqual[phi2, 3400000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.37 or 3.4e6 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]
   12. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       3. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       9. lower-sin.f64 25.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

   13. Applied rewrites 25.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       2. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       3. cos-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       4. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       6. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       8. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

       9. lower-sin.f64 29.8%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

   15. Applied rewrites 29.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \mathsf{fma}\left(\cos \lambda_1, \cos \lambda_2, \sin \lambda_1 \cdot \sin \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

   if -0.37 < phi2 < 3.4e6

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 60.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_0\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_2 \leq 3400000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_0)))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi2 -0.37)
     t_3
     (if (<= phi2 3400000.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_0);
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = t_3;
	} else if (phi2 <= 3400000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_0))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = t_3;
	elseif (phi2 <= 3400000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], t$95$3, If[LessEqual[phi2, 3400000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.37 or 3.4e6 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]

   if -0.37 < phi2 < 3.4e6

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 63.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   14. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}}^{2}\right)}}\right) \]

       3. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

   15. Applied rewrites 60.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_3 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-30}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;\phi_1 \leq 60:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (fma
           (cos (* -0.5 lambda2))
           (sin (* 0.5 lambda1))
           (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
          2.0))
        (t_1 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0)))
        (t_2 (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
        (t_3 (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))))
   (if (<= phi1 -6.8e-30)
     t_3
     (if (<= phi1 60.0)
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0);
	double t_1 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double t_2 = fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0));
	double t_3 = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	double tmp;
	if (phi1 <= -6.8e-30) {
		tmp = t_3;
	} else if (phi1 <= 60.0) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0
	t_1 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	t_2 = fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_3 = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))))
	tmp = 0.0
	if (phi1 <= -6.8e-30)
		tmp = t_3;
	elseif (phi1 <= 60.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -6.8e-30], t$95$3, If[LessEqual[phi1, 60.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.8000000000000006e-30 or 60 < phi1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if -6.8000000000000006e-30 < phi1 < 60

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.1%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := 0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{t\_1}\right) \cdot t\_1}}\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (cos phi1)
          (pow
           (fma
            (cos (* -0.5 lambda2))
            (sin (* 0.5 lambda1))
            (* (cos (* 0.5 lambda1)) (sin (* -0.5 lambda2))))
           2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (t_1 (- 0.5 (* 0.5 (cos (- phi2)))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_2) t_2))))
   (if (<= phi2 -0.37)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (- 0.5 (* (cos phi2) 0.5))
          (* (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi2))))
        (sqrt
         (-
          1.0
          (*
           (+
            1.0
            (/ (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))) t_1))
           t_1))))))
     (if (<= phi2 2e-33)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(cos(phi1), pow(fma(cos((-0.5 * lambda2)), sin((0.5 * lambda1)), (cos((0.5 * lambda1)) * sin((-0.5 * lambda2)))), 2.0), pow(sin((0.5 * phi1)), 2.0));
	double t_1 = 0.5 - (0.5 * cos(-phi2));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_2) * t_2);
	double tmp;
	if (phi2 <= -0.37) {
		tmp = R * (2.0 * atan2(sqrt(((0.5 - (cos(phi2) * 0.5)) + ((0.5 - (cos((lambda2 - lambda1)) * 0.5)) * cos(phi2)))), sqrt((1.0 - ((1.0 + ((cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))) / t_1)) * t_1)))));
	} else if (phi2 <= 2e-33) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(cos(phi1), (fma(cos(Float64(-0.5 * lambda2)), sin(Float64(0.5 * lambda1)), Float64(cos(Float64(0.5 * lambda1)) * sin(Float64(-0.5 * lambda2)))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_1 = Float64(0.5 - Float64(0.5 * cos(Float64(-phi2))))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)) * cos(phi2)))), sqrt(Float64(1.0 - Float64(Float64(1.0 + Float64(Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))) / t_1)) * t_1))))));
	elseif (phi2 <= 2e-33)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[(N[Cos[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] + N[(N[Cos[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(0.5 * N[Cos[(-phi2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(1.0 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2e-33], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.37

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

   if -0.37 < phi2 < 2.0000000000000001e-33

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 56.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right), \sin \left(\frac{1}{2} \cdot \lambda_1\right), \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\left(\cos \left(\frac{-1}{2} \cdot \lambda_2\right) \cdot \sin \left(\frac{1}{2} \cdot \lambda_1\right) + \cos \left(\frac{1}{2} \cdot \lambda_1\right) \cdot \sin \left(\frac{-1}{2} \cdot \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 56.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \lambda_2\right), \sin \left(0.5 \cdot \lambda_1\right), \cos \left(0.5 \cdot \lambda_1\right) \cdot \sin \left(-0.5 \cdot \lambda_2\right)\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   if 2.0000000000000001e-33 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := 0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\\ t_4 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}\\ t_5 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\\ \mathbf{if}\;t\_1 \leq -0.015:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_4}{\sqrt{-\mathsf{fma}\left(t\_5, t\_2, -t\_3\right)}}\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_4}{\sqrt{t\_3 - t\_5 \cdot t\_2}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2)))))))
        (t_4
         (sqrt
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* (* (* (cos phi1) (cos phi2)) t_1) t_1))))
        (t_5 (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))))
   (if (<= t_1 -0.015)
     (* R (* 2.0 (atan2 t_4 (sqrt (- (fma t_5 t_2 (- t_3)))))))
     (if (<= t_1 2e-22)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (* R (* 2.0 (atan2 t_4 (sqrt (- t_3 (* t_5 t_2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = 0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))));
	double t_4 = sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1)));
	double t_5 = 0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))));
	double tmp;
	if (t_1 <= -0.015) {
		tmp = R * (2.0 * atan2(t_4, sqrt(-fma(t_5, t_2, -t_3))));
	} else if (t_1 <= 2e-22) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(t_4, sqrt((t_3 - (t_5 * t_2)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2))))))
	t_4 = sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1)))
	t_5 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))
	tmp = 0.0
	if (t_1 <= -0.015)
		tmp = Float64(R * Float64(2.0 * atan(t_4, sqrt(Float64(-fma(t_5, t_2, Float64(-t_3)))))));
	elseif (t_1 <= 2e-22)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_4, sqrt(Float64(t_3 - Float64(t_5 * t_2))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.015], N[(R * N[(2.0 * N[ArcTan[t$95$4 / N[Sqrt[(-N[(t$95$5 * t$95$2 + (-t$95$3)), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-22], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$4 / N[Sqrt[N[(t$95$3 - N[(t$95$5 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.014999999999999999

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{-\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \cos \phi_2 \cdot \cos \phi_1, -\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\right)}}}\right) \]

   if -0.014999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2.0000000000000001e-22

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   22. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   23. Applied rewrites 29.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   24. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

   25. Applied rewrites 34.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}\right) \]

   if 2.0000000000000001e-22 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 64.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} t_0 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_1\right) \cdot t\_1}}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}\right)\\ \mathbf{if}\;t\_1 \leq -0.015:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-22}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (+
              (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
              (* (* (* (cos phi1) (cos phi2)) t_1) t_1)))
            (sqrt
             (-
              (+ 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))
              (*
               (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2))))))
               (* (cos phi2) (cos phi1))))))))))
   (if (<= t_1 -0.015)
     t_2
     (if (<= t_1 2e-22)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
	double tmp;
	if (t_1 <= -0.015) {
		tmp = t_2;
	} else if (t_1 <= 2e-22) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = ((sin((phi1 * 0.5d0)) * cos((0.5d0 * phi2))) - (cos((phi1 * 0.5d0)) * sin((0.5d0 * phi2)))) ** 2.0d0
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(((0.5d0 + (0.5d0 * cos((2.0d0 * (0.5d0 * (phi1 - phi2)))))) - ((0.5d0 - (0.5d0 * cos((2.0d0 * (0.5d0 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))))
    if (t_1 <= (-0.015d0)) then
        tmp = t_2
    else if (t_1 <= 2d-22) then
        tmp = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(((Math.sin((phi1 * 0.5)) * Math.cos((0.5 * phi2))) - (Math.cos((phi1 * 0.5)) * Math.sin((0.5 * phi2)))), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((Math.cos(phi1) * Math.cos(phi2)) * t_1) * t_1))), Math.sqrt(((0.5 + (0.5 * Math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * Math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (Math.cos(phi2) * Math.cos(phi1)))))));
	double tmp;
	if (t_1 <= -0.015) {
		tmp = t_2;
	} else if (t_1 <= 2e-22) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(((math.sin((phi1 * 0.5)) * math.cos((0.5 * phi2))) - (math.cos((phi1 * 0.5)) * math.sin((0.5 * phi2)))), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (((math.cos(phi1) * math.cos(phi2)) * t_1) * t_1))), math.sqrt(((0.5 + (0.5 * math.cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * math.cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (math.cos(phi2) * math.cos(phi1)))))))
	tmp = 0
	if t_1 <= -0.015:
		tmp = t_2
	elif t_1 <= 2e-22:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))
	else:
		tmp = t_2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))) - Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2)))))) * Float64(cos(phi2) * cos(phi1))))))))
	tmp = 0.0
	if (t_1 <= -0.015)
		tmp = t_2;
	elseif (t_1 <= 2e-22)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = ((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))) ^ 2.0;
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (((cos(phi1) * cos(phi2)) * t_1) * t_1))), sqrt(((0.5 + (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))) - ((0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2)))))) * (cos(phi2) * cos(phi1)))))));
	tmp = 0.0;
	if (t_1 <= -0.015)
		tmp = t_2;
	elseif (t_1 <= 2e-22)
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.015], t$95$2, If[LessEqual[t$95$1, 2e-22], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.014999999999999999 or 2.0000000000000001e-22 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\color{blue}{\sqrt{\left(0.5 + 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right) - \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}\right) \]

   if -0.014999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 2.0000000000000001e-22

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   22. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   23. Applied rewrites 29.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   24. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

   25. Applied rewrites 34.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_3 := t\_2 \cdot t\_0\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_3\\ t_5 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_6 := \cos \left(\phi_2 - \phi_1\right)\\ \mathbf{if}\;t\_1 \leq -0.015:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_0, \mathsf{fma}\left(t\_6, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_6 \cdot -0.5\right) - t\_3}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_3 (* t_2 t_0))
        (t_4 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_3))
        (t_5
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0))
        (t_6 (cos (- phi2 phi1))))
   (if (<= t_1 -0.015)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= t_1 5e-13)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (*
        (*
         (atan2
          (sqrt (fma t_2 t_0 (fma t_6 -0.5 0.5)))
          (sqrt (- (- 0.5 (* t_6 -0.5)) t_3)))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_3 = t_2 * t_0;
	double t_4 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_3;
	double t_5 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double t_6 = cos((phi2 - phi1));
	double tmp;
	if (t_1 <= -0.015) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (t_1 <= 5e-13) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = (atan2(sqrt(fma(t_2, t_0, fma(t_6, -0.5, 0.5))), sqrt(((0.5 - (t_6 * -0.5)) - t_3))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_3 = Float64(t_2 * t_0)
	t_4 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_3)
	t_5 = Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_6 = cos(Float64(phi2 - phi1))
	tmp = 0.0
	if (t_1 <= -0.015)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (t_1 <= 5e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_2, t_0, fma(t_6, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_6 * -0.5)) - t_3))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * t$95$0), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$6 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.015], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 5e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$0 + N[(t$95$6 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$6 * -0.5), $MachinePrecision]), $MachinePrecision] - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.014999999999999999

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 59.6%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Applied rewrites 59.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}\right)}}\right) \]

   if -0.014999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 4.9999999999999999e-13

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   22. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   23. Applied rewrites 29.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   24. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

   25. Applied rewrites 34.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}\right) \]

   if 4.9999999999999999e-13 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 56.6%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_1 := \mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := \cos \left(\phi_2 - \phi_1\right)\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_5 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ \mathbf{if}\;t\_4 \leq -0.015:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, t\_2, \mathsf{fma}\left(t\_3, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_3 \cdot -0.5\right) - t\_0 \cdot t\_2}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_1
         (fma
          (cos phi2)
          (*
           (cos phi1)
           (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- lambda1 lambda2)))))))
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 (- phi1 phi2))))))))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3 (cos (- phi2 phi1)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_5
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0)))
   (if (<= t_4 -0.015)
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (if (<= t_4 5e-13)
       (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
       (*
        (*
         (atan2
          (sqrt (fma t_0 t_2 (fma t_3 -0.5 0.5)))
          (sqrt (- (- 0.5 (* t_3 -0.5)) (* t_0 t_2))))
         2.0)
        R)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_1 = fma(cos(phi2), (cos(phi1) * (0.5 - (0.5 * cos((2.0 * (0.5 * (lambda1 - lambda2))))))), (0.5 - (0.5 * cos((2.0 * (0.5 * (phi1 - phi2)))))));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = cos((phi2 - phi1));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double t_5 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double tmp;
	if (t_4 <= -0.015) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else if (t_4 <= 5e-13) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = (atan2(sqrt(fma(t_0, t_2, fma(t_3, -0.5, 0.5))), sqrt(((0.5 - (t_3 * -0.5)) - (t_0 * t_2)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_1 = fma(cos(phi2), Float64(cos(phi1) * Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(lambda1 - lambda2))))))), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * Float64(phi1 - phi2)))))))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = cos(Float64(phi2 - phi1))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_5 = Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	tmp = 0.0
	if (t_4 <= -0.015)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	elseif (t_4 <= 5e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_0, t_2, fma(t_3, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_3 * -0.5)) - Float64(t_0 * t_2)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[t$95$4, -0.015], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 5e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$0 * t$95$2 + N[(t$95$3 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$3 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.014999999999999999

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 56.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right), 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   if -0.014999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 4.9999999999999999e-13

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   22. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   23. Applied rewrites 29.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   24. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

   25. Applied rewrites 34.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}\right) \]

   if 4.9999999999999999e-13 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 56.6%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 63.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_2 \cdot \cos \phi_1\\ t_3 := {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_4 := \cos \left(\phi_2 - \phi_1\right)\\ t_5 := \left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, t\_2, \mathsf{fma}\left(t\_4, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_4 \cdot -0.5\right) - t\_0 \cdot t\_2}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_1 \leq -0.015:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (cos phi2) (cos phi1)))
        (t_3
         (pow
          (-
           (* (sin (* phi1 0.5)) (cos (* 0.5 phi2)))
           (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
          2.0))
        (t_4 (cos (- phi2 phi1)))
        (t_5
         (*
          (*
           (atan2
            (sqrt (fma t_0 t_2 (fma t_4 -0.5 0.5)))
            (sqrt (- (- 0.5 (* t_4 -0.5)) (* t_0 t_2))))
           2.0)
          R)))
   (if (<= t_1 -0.015)
     t_5
     (if (<= t_1 5e-13)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_5))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos(phi2) * cos(phi1);
	double t_3 = pow(((sin((phi1 * 0.5)) * cos((0.5 * phi2))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	double t_4 = cos((phi2 - phi1));
	double t_5 = (atan2(sqrt(fma(t_0, t_2, fma(t_4, -0.5, 0.5))), sqrt(((0.5 - (t_4 * -0.5)) - (t_0 * t_2)))) * 2.0) * R;
	double tmp;
	if (t_1 <= -0.015) {
		tmp = t_5;
	} else if (t_1 <= 5e-13) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(cos(phi2) * cos(phi1))
	t_3 = Float64(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_4 = cos(Float64(phi2 - phi1))
	t_5 = Float64(Float64(atan(sqrt(fma(t_0, t_2, fma(t_4, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_4 * -0.5)) - Float64(t_0 * t_2)))) * 2.0) * R)
	tmp = 0.0
	if (t_1 <= -0.015)
		tmp = t_5;
	elseif (t_1 <= 5e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[(N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[ArcTan[N[Sqrt[N[(t$95$0 * t$95$2 + N[(t$95$4 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$4 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$0 * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$1, -0.015], t$95$5, If[LessEqual[t$95$1, 5e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$5]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.014999999999999999 or 4.9999999999999999e-13 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 56.6%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]

   if -0.014999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 4.9999999999999999e-13

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   22. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   23. Applied rewrites 29.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   24. Step-by-step derivation

       1. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       4. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot \frac{1}{2}\right)}^{2}}}\right) \]

       5. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       6. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}\right) \]

       7. div-sub N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{\phi_1}{2} - \frac{\phi_2}{2}\right)}^{2}}}\right) \]

       8. sin-diff N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       9. lower--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       10. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       11. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       12. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       13. lift-sin.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       14. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       15. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       16. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       17. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       18. lower-cos.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       19. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\phi_2 \cdot \frac{1}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       20. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       21. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}}}\right) \]

       22. mult-flip-rev N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       23. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

       24. lift-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{1}{2} \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \cos \left(\frac{1}{2} \cdot \phi_2\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\phi_2 \cdot \frac{1}{2}\right)\right)}^{2}}}\right) \]

   25. Applied rewrites 34.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 62.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot t\_1\\ t_3 := \mathsf{fma}\left(\cos \phi_2, t\_1, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{t\_0}\right) \cdot t\_0}}\right)\\ \mathbf{elif}\;\phi_2 \leq 3400000:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (- phi2)))))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (cos phi1) t_1)))
        (t_3 (fma (cos phi2) t_1 (pow (sin (* -0.5 phi2)) 2.0))))
   (if (<= phi2 -0.37)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (- 0.5 (* (cos phi2) 0.5))
          (* (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi2))))
        (sqrt
         (-
          1.0
          (*
           (+
            1.0
            (/ (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))) t_0))
           t_0))))))
     (if (<= phi2 3400000.0)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos(-phi2));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (cos(phi1) * t_1);
	double t_3 = fma(cos(phi2), t_1, pow(sin((-0.5 * phi2)), 2.0));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = R * (2.0 * atan2(sqrt(((0.5 - (cos(phi2) * 0.5)) + ((0.5 - (cos((lambda2 - lambda1)) * 0.5)) * cos(phi2)))), sqrt((1.0 - ((1.0 + ((cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))) / t_0)) * t_0)))));
	} else if (phi2 <= 3400000.0) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(-phi2))))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(cos(phi1) * t_1))
	t_3 = fma(cos(phi2), t_1, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)) * cos(phi2)))), sqrt(Float64(1.0 - Float64(Float64(1.0 + Float64(Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))) / t_0)) * t_0))))));
	elseif (phi2 <= 3400000.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[(-phi2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(1.0 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3400000.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -0.37

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

   if -0.37 < phi2 < 3.4e6

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 3.4e6 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 20: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \cos \phi_2 \cdot \cos \phi_1\\ t_1 := \cos \left(\phi_2 - \phi_1\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right)\\ t_5 := t\_3 + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;t\_3 + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_2\right) \cdot t\_2 \leq 0.008:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_5}}{\sqrt{1 - t\_5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, t\_0, \mathsf{fma}\left(t\_1, -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - t\_1 \cdot -0.5\right) - t\_4 \cdot t\_0}} \cdot 2\right) \cdot R\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (cos phi1)))
        (t_1 (cos (- phi2 phi1)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (fma -0.5 (cos (- lambda2 lambda1)) 0.5))
        (t_5
         (+ t_3 (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))))
   (if (<= (+ t_3 (* (* (* (cos phi1) (cos phi2)) t_2) t_2)) 0.008)
     (* R (* 2.0 (atan2 (sqrt t_5) (sqrt (- 1.0 t_5)))))
     (*
      (*
       (atan2
        (sqrt (fma t_4 t_0 (fma t_1 -0.5 0.5)))
        (sqrt (- (- 0.5 (* t_1 -0.5)) (* t_4 t_0))))
       2.0)
      R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * cos(phi1);
	double t_1 = cos((phi2 - phi1));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = fma(-0.5, cos((lambda2 - lambda1)), 0.5);
	double t_5 = t_3 + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	double tmp;
	if ((t_3 + (((cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 0.008) {
		tmp = R * (2.0 * atan2(sqrt(t_5), sqrt((1.0 - t_5))));
	} else {
		tmp = (atan2(sqrt(fma(t_4, t_0, fma(t_1, -0.5, 0.5))), sqrt(((0.5 - (t_1 * -0.5)) - (t_4 * t_0)))) * 2.0) * R;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * cos(phi1))
	t_1 = cos(Float64(phi2 - phi1))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5)
	t_5 = Float64(t_3 + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))
	tmp = 0.0
	if (Float64(t_3 + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_2) * t_2)) <= 0.008)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_5), sqrt(Float64(1.0 - t_5)))));
	else
		tmp = Float64(Float64(atan(sqrt(fma(t_4, t_0, fma(t_1, -0.5, 0.5))), sqrt(Float64(Float64(0.5 - Float64(t_1 * -0.5)) - Float64(t_4 * t_0)))) * 2.0) * R);
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(phi2 - phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$3 + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 0.008], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$5], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[ArcTan[N[Sqrt[N[(t$95$4 * t$95$0 + N[(t$95$1 * -0.5 + 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(0.5 - N[(t$95$1 * -0.5), $MachinePrecision]), $MachinePrecision] - N[(t$95$4 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))))) < 0.0080000000000000002

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 52.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 52.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

      6. lower--.f64 50.6%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 50.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}\right)}}\right) \]

   if 0.0080000000000000002 < (+.f64 (pow.f64 (sin.f64 (/.f64 (-.f64 phi1 phi2) #s(literal 2 binary64))) #s(literal 2 binary64)) (*.f64 (*.f64 (*.f64 (cos.f64 phi1) (cos.f64 phi2)) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))) (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   11. Applied rewrites 56.6%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_2 \cdot \cos \phi_1, \mathsf{fma}\left(\cos \left(\phi_2 - \phi_1\right), -0.5, 0.5\right)\right)}}{\sqrt{\left(0.5 - \cos \left(\phi_2 - \phi_1\right) \cdot -0.5\right) - \mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 21: 61.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_0, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ t_2 := \mathsf{fma}\left(\cos \phi_2, t\_0, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -2 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_1 \leq 60:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1
         (*
          R
          (*
           2.0
           (atan2
            (sqrt (fma (cos phi1) t_0 (pow (sin (* 0.5 phi1)) 2.0)))
            (sqrt
             (-
              1.0
              (fma
               (fma -0.5 (cos (- lambda2 lambda1)) 0.5)
               (cos phi1)
               (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))))))))
        (t_2 (fma (cos phi2) t_0 (pow (sin (* -0.5 phi2)) 2.0))))
   (if (<= phi1 -2e-7)
     t_1
     (if (<= phi1 60.0)
       (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_0, pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, cos((lambda2 - lambda1)), 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	double t_2 = fma(cos(phi2), t_0, pow(sin((-0.5 * phi2)), 2.0));
	double tmp;
	if (phi1 <= -2e-7) {
		tmp = t_1;
	} else if (phi1 <= 60.0) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_0, (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, cos(Float64(lambda2 - lambda1)), 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))))
	t_2 = fma(cos(phi2), t_0, (sin(Float64(-0.5 * phi2)) ^ 2.0))
	tmp = 0.0
	if (phi1 <= -2e-7)
		tmp = t_1;
	elseif (phi1 <= 60.0)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -2e-7], t$95$1, If[LessEqual[phi1, 60.0], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.9999999999999999e-7 or 60 < phi1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]

   if -1.9999999999999999e-7 < phi1 < 60

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_2 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{-1}{2} \cdot \phi_2\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.5%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_2, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 22: 60.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\\ t_1 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - t\_1 \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{t\_0}\right) \cdot t\_0}}\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_1, 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (- phi2)))))
        (t_1 (cos (- lambda2 lambda1)))
        (t_2
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (+ (- 0.5 (* (cos phi2) 0.5)) (* (- 0.5 (* t_1 0.5)) (cos phi2))))
            (sqrt
             (-
              1.0
              (*
               (+
                1.0
                (/
                 (* (cos phi2) (- 0.5 (* 0.5 (cos (- lambda1 lambda2)))))
                 t_0))
               t_0))))))))
   (if (<= phi2 -0.37)
     t_2
     (if (<= phi2 1.55e+28)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 t_1 0.5)
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos(-phi2));
	double t_1 = cos((lambda2 - lambda1));
	double t_2 = R * (2.0 * atan2(sqrt(((0.5 - (cos(phi2) * 0.5)) + ((0.5 - (t_1 * 0.5)) * cos(phi2)))), sqrt((1.0 - ((1.0 + ((cos(phi2) * (0.5 - (0.5 * cos((lambda1 - lambda2))))) / t_0)) * t_0)))));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = t_2;
	} else if (phi2 <= 1.55e+28) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, t_1, 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(-phi2))))
	t_1 = cos(Float64(lambda2 - lambda1))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(t_1 * 0.5)) * cos(phi2)))), sqrt(Float64(1.0 - Float64(Float64(1.0 + Float64(Float64(cos(phi2) * Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2))))) / t_0)) * t_0))))))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = t_2;
	elseif (phi2 <= 1.55e+28)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, t_1, 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[(-phi2)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(t$95$1 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(1.0 + N[(N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], t$95$2, If[LessEqual[phi2, 1.55e+28], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * t$95$1 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.37 or 1.55e28 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}\right) \]

   if -0.37 < phi2 < 1.55e28

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 23: 60.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - t\_0 \cdot 0.5\right) \cdot \cos \phi_2\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_0, 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (+ (- 0.5 (* (cos phi2) 0.5)) (* (- 0.5 (* t_0 0.5)) (cos phi2))))
        (t_2 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R)))
   (if (<= phi2 -0.37)
     t_2
     (if (<= phi2 1.55e+28)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (pow (sin (* 0.5 phi1)) 2.0)))
          (sqrt
           (-
            1.0
            (fma
             (fma -0.5 t_0 0.5)
             (cos phi1)
             (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1)))))))))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = (0.5 - (cos(phi2) * 0.5)) + ((0.5 - (t_0 * 0.5)) * cos(phi2));
	double t_2 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double tmp;
	if (phi2 <= -0.37) {
		tmp = t_2;
	} else if (phi2 <= 1.55e+28) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((1.0 - fma(fma(-0.5, t_0, 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))))))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(t_0 * 0.5)) * cos(phi2)))
	t_2 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = t_2;
	elseif (phi2 <= 1.55e+28)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(1.0 - fma(fma(-0.5, t_0, 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1)))))))))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi2, -0.37], t$95$2, If[LessEqual[phi2, 1.55e+28], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.37 or 1.55e28 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R} \]

   if -0.37 < phi2 < 1.55e28

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\color{blue}{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 24: 58.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - t\_0 \cdot 0.5\right) \cdot \cos \phi_2\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ t_3 := \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(t\_0, -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (+ (- 0.5 (* (cos phi2) 0.5)) (* (- 0.5 (* t_0 0.5)) (cos phi2))))
        (t_2 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R))
        (t_3 (fma (cos phi1) (fma t_0 -0.5 0.5) (pow (sin (* 0.5 phi1)) 2.0))))
   (if (<= phi2 -0.37)
     t_2
     (if (<= phi2 1.55e+28)
       (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = (0.5 - (cos(phi2) * 0.5)) + ((0.5 - (t_0 * 0.5)) * cos(phi2));
	double t_2 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double t_3 = fma(cos(phi1), fma(t_0, -0.5, 0.5), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = t_2;
	} else if (phi2 <= 1.55e+28) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(t_0 * 0.5)) * cos(phi2)))
	t_2 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	t_3 = fma(cos(phi1), fma(t_0, -0.5, 0.5), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = t_2;
	elseif (phi2 <= 1.55e+28)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[(t$95$0 * -0.5 + 0.5), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], t$95$2, If[LessEqual[phi2, 1.55e+28], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.37 or 1.55e28 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R} \]

   if -0.37 < phi2 < 1.55e28

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   8. Step-by-step derivation

      1. lift-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. unpow2 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. sqr-sin-a-rev N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lift-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. +-commutative N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      11. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{-1}{2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      12. lower-fma.f64 43.7%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   9. Applied rewrites 43.7%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   10. Step-by-step derivation

       1. lift-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. unpow2 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \color{blue}{\left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       4. lift-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       5. sqr-sin-a-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \color{blue}{\frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       6. lift-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       7. lift-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       8. fp-cancel-sub-sign-inv N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{1}{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       9. +-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \color{blue}{\frac{1}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       10. metadata-eval N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \frac{-1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       11. *-commutative N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \frac{-1}{2}, \frac{1}{2}\right), {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right) \cdot \frac{-1}{2} + \frac{1}{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       12. lower-fma.f64 43.7%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Applied rewrites 43.7%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), -0.5, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \mathsf{fma}\left(\cos \left(\lambda_2 - \lambda_1\right), \color{blue}{-0.5}, 0.5\right), {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 25: 56.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\lambda_2 - \lambda_1\right)\\ t_1 := \left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - t\_0 \cdot 0.5\right) \cdot \cos \phi_2\\ t_2 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ t_3 := \mathsf{fma}\left(\mathsf{fma}\left(-0.5, t\_0, 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)\\ \mathbf{if}\;\phi_2 \leq -0.37:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;\phi_2 \leq 1.55 \cdot 10^{+28}:\\ \;\;\;\;\left(\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot 2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (- lambda2 lambda1)))
        (t_1 (+ (- 0.5 (* (cos phi2) 0.5)) (* (- 0.5 (* t_0 0.5)) (cos phi2))))
        (t_2 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R))
        (t_3
         (fma
          (fma -0.5 t_0 0.5)
          (cos phi1)
          (- 0.5 (* 0.5 (cos (* 2.0 (* 0.5 phi1))))))))
   (if (<= phi2 -0.37)
     t_2
     (if (<= phi2 1.55e+28)
       (* (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) 2.0) R)
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((lambda2 - lambda1));
	double t_1 = (0.5 - (cos(phi2) * 0.5)) + ((0.5 - (t_0 * 0.5)) * cos(phi2));
	double t_2 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double t_3 = fma(fma(-0.5, t_0, 0.5), cos(phi1), (0.5 - (0.5 * cos((2.0 * (0.5 * phi1))))));
	double tmp;
	if (phi2 <= -0.37) {
		tmp = t_2;
	} else if (phi2 <= 1.55e+28) {
		tmp = (atan2(sqrt(t_3), sqrt((1.0 - t_3))) * 2.0) * R;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(lambda2 - lambda1))
	t_1 = Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(t_0 * 0.5)) * cos(phi2)))
	t_2 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	t_3 = fma(fma(-0.5, t_0, 0.5), cos(phi1), Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(0.5 * phi1))))))
	tmp = 0.0
	if (phi2 <= -0.37)
		tmp = t_2;
	elseif (phi2 <= 1.55e+28)
		tmp = Float64(Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * 2.0) * R);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(t$95$0 * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, Block[{t$95$3 = N[(N[(-0.5 * t$95$0 + 0.5), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(0.5 * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, -0.37], t$95$2, If[LessEqual[phi2, 1.55e+28], N[(N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.37 or 1.55e28 < phi2

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R} \]

   if -0.37 < phi2 < 1.55e28

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   9. Applied rewrites 42.2%

      \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\mathsf{fma}\left(-0.5, \cos \left(\lambda_2 - \lambda_1\right), 0.5\right), \cos \phi_1, 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \phi_1\right)\right)\right)}} \cdot 2\right) \cdot R} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 26: 51.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_1 := \left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \left(\tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}} \cdot 2\right) \cdot R\\ \mathbf{if}\;t\_2 \leq -0.015:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, t\_0\right)}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_1
         (+
          (- 0.5 (* (cos phi2) 0.5))
          (* (- 0.5 (* (cos (- lambda2 lambda1)) 0.5)) (cos phi2))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* (* (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))) 2.0) R)))
   (if (<= t_2 -0.015)
     t_3
     (if (<= t_2 5e-13)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) (pow (sin (* -0.5 lambda2)) 2.0))
            t_0))
          (sqrt (- 1.0 t_0)))))
       t_3))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_1 = (0.5 - (cos(phi2) * 0.5)) + ((0.5 - (cos((lambda2 - lambda1)) * 0.5)) * cos(phi2));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = (atan2(sqrt(t_1), sqrt((1.0 - t_1))) * 2.0) * R;
	double tmp;
	if (t_2 <= -0.015) {
		tmp = t_3;
	} else if (t_2 <= 5e-13) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin((-0.5 * lambda2)), 2.0)), t_0)), sqrt((1.0 - t_0))));
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_1 = Float64(Float64(0.5 - Float64(cos(phi2) * 0.5)) + Float64(Float64(0.5 - Float64(cos(Float64(lambda2 - lambda1)) * 0.5)) * cos(phi2)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(Float64(atan(sqrt(t_1), sqrt(Float64(1.0 - t_1))) * 2.0) * R)
	tmp = 0.0
	if (t_2 <= -0.015)
		tmp = t_3;
	elseif (t_2 <= 5e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(-0.5 * lambda2)) ^ 2.0)), t_0)), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(0.5 - N[(N[Cos[phi2], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 - N[(N[Cos[N[(lambda2 - lambda1), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[t$95$2, -0.015], t$95$3, If[LessEqual[t$95$2, 5e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.014999999999999999 or 4.9999999999999999e-13 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 43.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}}\right) \]

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   8. Applied rewrites 24.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}{\sqrt{1 - \left(1 + \frac{\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right) \cdot \left(\cos \phi_2 \cdot \cos \phi_1\right)}{0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\right)\right)}}\right) \]

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]
   10. Step-by-step derivation

       1. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \left(1 + \frac{\cos \phi_2 \cdot \left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(\mathsf{neg}\left(\phi_2\right)\right)\right)}}}\right) \]

   11. Applied rewrites 25.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}{\sqrt{1 - \color{blue}{\left(1 + \frac{\cos \phi_2 \cdot \left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)}{0.5 - 0.5 \cdot \cos \left(-\phi_2\right)}\right) \cdot \left(0.5 - 0.5 \cdot \cos \left(-\phi_2\right)\right)}}}\right) \]

   13. Applied rewrites 42.5%

       \[\leadsto \color{blue}{\left(\tan^{-1}_* \frac{\sqrt{\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2}}{\sqrt{1 - \left(\left(0.5 - \cos \phi_2 \cdot 0.5\right) + \left(0.5 - \cos \left(\lambda_2 - \lambda_1\right) \cdot 0.5\right) \cdot \cos \phi_2\right)}} \cdot 2\right) \cdot R} \]

   if -0.014999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 4.9999999999999999e-13

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   22. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   23. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   24. Applied rewrites 32.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 27: 38.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}\right)\\ \mathbf{if}\;t\_1 \leq -0.03:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.38:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, t\_0\right)}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (fma
              (cos phi1)
              (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
              (pow (* 0.5 phi1) 2.0)))
            (pow
             (-
              1.0
              (fma
               (* phi1 0.5)
               (* phi1 0.5)
               (*
                (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
                (cos phi1))))
             0.5))))))
   (if (<= t_1 -0.03)
     t_2
     (if (<= t_1 0.38)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) (pow (sin (* -0.5 lambda2)) 2.0))
            t_0))
          (sqrt (- 1.0 t_0)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow((0.5 * phi1), 2.0))), pow((1.0 - fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)))), 0.5)));
	double tmp;
	if (t_1 <= -0.03) {
		tmp = t_2;
	} else if (t_1 <= 0.38) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin((-0.5 * lambda2)), 2.0)), t_0)), sqrt((1.0 - t_0))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (Float64(0.5 * phi1) ^ 2.0))), (Float64(1.0 - fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))) ^ 0.5))))
	tmp = 0.0
	if (t_1 <= -0.03)
		tmp = t_2;
	elseif (t_1 <= 0.38)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(-0.5 * lambda2)) ^ 2.0)), t_0)), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.03], t$95$2, If[LessEqual[t$95$1, 0.38], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.029999999999999999 or 0.38 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 27.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}}\right) \]

   if -0.029999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.38

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   22. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   23. Step-by-step derivation

       1. lower-sqrt.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       6. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       7. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       8. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       9. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   24. Applied rewrites 32.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\color{blue}{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 28: 36.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}\right)\\ \mathbf{if}\;t\_0 \leq -0.03:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0.38:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (*
          R
          (*
           2.0
           (atan2
            (sqrt
             (fma
              (cos phi1)
              (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
              (pow (* 0.5 phi1) 2.0)))
            (pow
             (-
              1.0
              (fma
               (* phi1 0.5)
               (* phi1 0.5)
               (*
                (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
                (cos phi1))))
             0.5))))))
   (if (<= t_0 -0.03)
     t_1
     (if (<= t_0 0.38)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
          (exp
           (*
            (log (- 1.0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5))))
            0.5)))))
       t_1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow((0.5 * phi1), 2.0))), pow((1.0 - fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)))), 0.5)));
	double tmp;
	if (t_0 <= -0.03) {
		tmp = t_1;
	} else if (t_0 <= 0.38) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin((0.5 * (phi1 - phi2))), 2.0)), exp((log((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (Float64(0.5 * phi1) ^ 2.0))), (Float64(1.0 - fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))) ^ 0.5))))
	tmp = 0.0
	if (t_0 <= -0.03)
		tmp = t_1;
	elseif (t_0 <= 0.38)
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)), exp(Float64(log(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5)))) * 0.5)))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Power[N[(1.0 - N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.03], t$95$1, If[LessEqual[t$95$0, 0.38], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.029999999999999999 or 0.38 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 27.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\color{blue}{{\left(1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\right)}^{0.5}}}\right) \]

   if -0.029999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.38

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   23. Applied rewrites 29.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\color{blue}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 29: 36.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_1 := {\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.00012:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}\right)\\ \mathbf{elif}\;\phi_1 \leq 4.8 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (+ 1.0 (* -0.5 (pow phi1 2.0)))
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (* 0.5 phi1) 2.0)))
        (t_1 (pow (sin (* 0.5 (* (- 1.0 (/ phi2 phi1)) phi1))) 2.0)))
   (if (<= phi1 -0.00012)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (exp
         (* (log (- 1.0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5)))) 0.5)))))
     (if (<= phi1 4.8e-9)
       (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
       (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((1.0 + (-0.5 * pow(phi1, 2.0))), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow((0.5 * phi1), 2.0));
	double t_1 = pow(sin((0.5 * ((1.0 - (phi2 / phi1)) * phi1))), 2.0);
	double tmp;
	if (phi1 <= -0.00012) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin((0.5 * (phi1 - phi2))), 2.0)), exp((log((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
	} else if (phi1 <= 4.8e-9) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(1.0 + Float64(-0.5 * (phi1 ^ 2.0))), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (Float64(0.5 * phi1) ^ 2.0))
	t_1 = sin(Float64(0.5 * Float64(Float64(1.0 - Float64(phi2 / phi1)) * phi1))) ^ 2.0
	tmp = 0.0
	if (phi1 <= -0.00012)
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)), exp(Float64(log(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5)))) * 0.5)))));
	elseif (phi1 <= 4.8e-9)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(1.0 + N[(-0.5 * N[Power[phi1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(0.5 * phi1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(N[(1.0 - N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] * phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[phi1, -0.00012], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 4.8e-9], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.2e-4

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   23. Applied rewrites 29.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\color{blue}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}}\right) \]

   if -1.2e-4 < phi1 < 4.8e-9

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   14. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   15. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lower-pow.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   16. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   17. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   18. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + \frac{-1}{2} \cdot {\phi_1}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lower-pow.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   19. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(1 + -0.5 \cdot {\phi_1}^{2}, {\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   if 4.8e-9 < phi1

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   22. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. sub-to-mult N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-unsound-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower-unsound--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       5. lower-unsound-/.f64 23.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   23. Applied rewrites 23.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]
   24. Step-by-step derivation

       1. lift--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. sub-to-mult N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}\right) \]

       3. lower-unsound-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}\right) \]

       4. lower-unsound--.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}\right) \]

       5. lower-unsound-/.f64 23.9%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}\right) \]

   25. Applied rewrites 23.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\left(1 - \frac{\phi_2}{\phi_1}\right) \cdot \phi_1\right)\right)}^{2}}}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 30: 32.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, t\_0 \cdot \cos \phi_1\right)\\ \mathbf{if}\;t\_2 \leq -0.03:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{elif}\;t\_2 \leq 0.38:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}} \cdot \left(2 \cdot R\right)\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5))))))
        (t_1 (fma t_0 (cos phi1) (* (* phi1 0.5) (* phi1 0.5))))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (fma (* phi1 0.5) (* phi1 0.5) (* t_0 (cos phi1)))))
   (if (<= t_2 -0.03)
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (if (<= t_2 0.38)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
          (exp
           (*
            (log (- 1.0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5))))
            0.5)))))
       (* (atan2 (sqrt t_3) (sqrt (- 1.0 t_3))) (* 2.0 R))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))));
	double t_1 = fma(t_0, cos(phi1), ((phi1 * 0.5) * (phi1 * 0.5)));
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = fma((phi1 * 0.5), (phi1 * 0.5), (t_0 * cos(phi1)));
	double tmp;
	if (t_2 <= -0.03) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else if (t_2 <= 0.38) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin((0.5 * (phi1 - phi2))), 2.0)), exp((log((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
	} else {
		tmp = atan2(sqrt(t_3), sqrt((1.0 - t_3))) * (2.0 * R);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5)))))
	t_1 = fma(t_0, cos(phi1), Float64(Float64(phi1 * 0.5) * Float64(phi1 * 0.5)))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(t_0 * cos(phi1)))
	tmp = 0.0
	if (t_2 <= -0.03)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	elseif (t_2 <= 0.38)
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)), exp(Float64(log(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5)))) * 0.5)))));
	else
		tmp = Float64(atan(sqrt(t_3), sqrt(Float64(1.0 - t_3))) * Float64(2.0 * R));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.03], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.38], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.029999999999999999

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   14. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \color{blue}{{\left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

       3. lower-fma.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \color{blue}{\cos \phi_1}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   16. Step-by-step derivation

       1. lift-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + \color{blue}{{\left(\frac{1}{2} \cdot \phi_1\right)}^{2}}\right)}}\right) \]

       2. *-commutative N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\frac{1}{2} - \frac{1}{2} \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \frac{1}{2}\right)\right), \cos \phi_1, \left(\phi_1 \cdot \frac{1}{2}\right) \cdot \left(\phi_1 \cdot \frac{1}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1 + {\color{blue}{\left(\frac{1}{2} \cdot \phi_1\right)}}^{2}\right)}}\right) \]

       3. lower-fma.f64 19.5%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \color{blue}{\cos \phi_1}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   17. Applied rewrites 19.5%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \cos \phi_1, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right), \color{blue}{\cos \phi_1}, \left(\phi_1 \cdot 0.5\right) \cdot \left(\phi_1 \cdot 0.5\right)\right)}}\right) \]

   if -0.029999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.38

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   23. Applied rewrites 29.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\color{blue}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}}\right) \]

   if 0.38 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 31: 32.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \cdot \left(2 \cdot R\right)\\ \mathbf{if}\;t\_1 \leq -0.03:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.38:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (fma
          (* phi1 0.5)
          (* phi1 0.5)
          (*
           (- 0.5 (* 0.5 (cos (* 2.0 (* (- lambda1 lambda2) 0.5)))))
           (cos phi1))))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))) (* 2.0 R))))
   (if (<= t_1 -0.03)
     t_2
     (if (<= t_1 0.38)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
          (exp
           (*
            (log (- 1.0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5))))
            0.5)))))
       t_2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = fma((phi1 * 0.5), (phi1 * 0.5), ((0.5 - (0.5 * cos((2.0 * ((lambda1 - lambda2) * 0.5))))) * cos(phi1)));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = atan2(sqrt(t_0), sqrt((1.0 - t_0))) * (2.0 * R);
	double tmp;
	if (t_1 <= -0.03) {
		tmp = t_2;
	} else if (t_1 <= 0.38) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin((0.5 * (phi1 - phi2))), 2.0)), exp((log((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = fma(Float64(phi1 * 0.5), Float64(phi1 * 0.5), Float64(Float64(0.5 - Float64(0.5 * cos(Float64(2.0 * Float64(Float64(lambda1 - lambda2) * 0.5))))) * cos(phi1)))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))) * Float64(2.0 * R))
	tmp = 0.0
	if (t_1 <= -0.03)
		tmp = t_2;
	elseif (t_1 <= 0.38)
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)), exp(Float64(log(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5)))) * 0.5)))));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi1 * 0.5), $MachinePrecision] * N[(phi1 * 0.5), $MachinePrecision] + N[(N[(0.5 - N[(0.5 * N[Cos[N[(2.0 * N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(2.0 * R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.03], t$95$2, If[LessEqual[t$95$1, 0.38], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.029999999999999999 or 0.38 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. lower-*.f64 46.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. Applied rewrites 46.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}}\right) \]
   6. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      2. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\color{blue}{\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      3. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      4. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      5. lower-*.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      6. lower--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      7. lower-pow.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(\frac{1}{2} \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]

      9. lower-*.f64 46.4%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   7. Applied rewrites 46.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}}\right) \]

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]
   9. Step-by-step derivation

      1. lower-*.f64 31.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   10. Applied rewrites 31.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   11. Taylor expanded in phi1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(\frac{1}{2} \cdot \phi_1\right)}^{2}\right)}}\right) \]
   12. Step-by-step derivation

       1. lower-*.f64 22.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   13. Applied rewrites 22.2%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 19.5%

       \[\leadsto \color{blue}{\tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}}{\sqrt{1 - \mathsf{fma}\left(\phi_1 \cdot 0.5, \phi_1 \cdot 0.5, \left(0.5 - 0.5 \cdot \cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)\right) \cdot \cos \phi_1\right)}} \cdot \left(2 \cdot R\right)} \]

   if -0.029999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 0.38

   1. Initial program 61.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   2. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. Applied rewrites 60.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   4. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. Applied rewrites 62.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   6. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. Applied rewrites 61.9%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   8. Step-by-step derivation

      1. lift-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      2. lift-/.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

      3. lift--.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

      4. sub-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

      5. div-add N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

      6. sin-sum N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

      8. lower-sin.f64 N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      9. mult-flip N/A

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      10. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      11. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      12. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      13. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      14. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      15. frac-2neg-rev N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      16. lower-/.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      18. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      19. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      20. mult-flip N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      21. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      22. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

      23. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

      24. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

      25. metadata-eval N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

   9. Applied rewrites 76.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

   10. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
   11. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

       11. lower--.f64 46.4%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   12. Applied rewrites 46.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

   13. Taylor expanded in lambda1 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
   14. Step-by-step derivation

       1. lower-fma.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower-cos.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       5. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       6. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       7. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       8. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       9. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       10. lower-*.f64 N/A

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       11. lower--.f64 45.9%

           \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   15. Applied rewrites 45.9%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

   16. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
   17. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   18. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

   19. Taylor expanded in lambda2 around 0

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
   20. Step-by-step derivation

       1. lower-pow.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       2. lower-sin.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       3. lower-*.f64 N/A

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

       4. lower--.f64 29.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

   21. Applied rewrites 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

   23. Applied rewrites 29.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\color{blue}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 32: 29.4% accurate, 3.5× speedup

Math

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)}}\right) \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
    (sqrt (- 1.0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(pow(sin((0.5 * (phi1 - phi2))), 2.0)), sqrt((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5))))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (2.0d0 * atan2(sqrt((sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)), sqrt((1.0d0 - (0.5d0 - (cos((1.0d0 * (phi1 - phi2))) * 0.5d0))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * Math.atan2(Math.sqrt(Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)), Math.sqrt((1.0 - (0.5 - (Math.cos((1.0 * (phi1 - phi2))) * 0.5))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * (2.0 * math.atan2(math.sqrt(math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)), math.sqrt((1.0 - (0.5 - (math.cos((1.0 * (phi1 - phi2))) * 0.5))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)), sqrt(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5)))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (2.0 * atan2(sqrt((sin((0.5 * (phi1 - phi2))) ^ 2.0)), sqrt((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 60.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 62.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 61.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

9. Applied rewrites 76.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

10. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
11. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lower--.f64 46.4%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

12. Applied rewrites 46.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

13. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
14. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    6. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    11. lower--.f64 45.9%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

15. Applied rewrites 45.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

16. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
17. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower--.f64 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

18. Applied rewrites 29.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
20. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    4. lower--.f64 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

21. Applied rewrites 29.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
22. Step-by-step derivation

    1. lift-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    2. unpow2 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}\right) \]

    3. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}\right) \]

    4. lift-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right) \cdot \sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}}\right) \]

23. Applied rewrites 29.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{\color{blue}{1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)}}}\right) \]
24. Add Preprocessing

Alternative 33: 29.4% accurate, 3.1× speedup

Math

\[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}\right) \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (*
   2.0
   (atan2
    (sqrt (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
    (exp (* (log (- 1.0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5)))) 0.5))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(pow(sin((0.5 * (phi1 - phi2))), 2.0)), exp((log((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (2.0d0 * atan2(sqrt((sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)), exp((log((1.0d0 - (0.5d0 - (cos((1.0d0 * (phi1 - phi2))) * 0.5d0)))) * 0.5d0))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * Math.atan2(Math.sqrt(Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)), Math.exp((Math.log((1.0 - (0.5 - (Math.cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * (2.0 * math.atan2(math.sqrt(math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)), math.exp((math.log((1.0 - (0.5 - (math.cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0)), exp(Float64(log(Float64(1.0 - Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5)))) * 0.5)))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * (2.0 * atan2(sqrt((sin((0.5 * (phi1 - phi2))) ^ 2.0)), exp((log((1.0 - (0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5)))) * 0.5))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Exp[N[(N[Log[N[(1.0 - N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.8%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 60.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 62.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 61.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

9. Applied rewrites 76.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

10. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
11. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lower--.f64 46.4%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

12. Applied rewrites 46.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

13. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
14. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    6. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    11. lower--.f64 45.9%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

15. Applied rewrites 45.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

16. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
17. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower--.f64 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

18. Applied rewrites 29.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
20. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    4. lower--.f64 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

21. Applied rewrites 29.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]

23. Applied rewrites 29.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\color{blue}{e^{\log \left(1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)\right) \cdot 0.5}}}\right) \]
24. Add Preprocessing

Alternative 34: 26.3% accurate, 3.8× speedup

Math

\[\begin{array}{l} t_0 := 0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\\ \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{t\_0}}{\sqrt{1 - t\_0}} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 0.5 (* (cos (* 1.0 (- phi1 phi2))) 0.5))))
   (* (* R 2.0) (atan2 (sqrt t_0) (sqrt (- 1.0 t_0))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5);
	return (R * 2.0) * atan2(sqrt(t_0), sqrt((1.0 - t_0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(r, lambda1, lambda2, phi1, phi2)
use fmin_fmax_functions
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    t_0 = 0.5d0 - (cos((1.0d0 * (phi1 - phi2))) * 0.5d0)
    code = (r * 2.0d0) * atan2(sqrt(t_0), sqrt((1.0d0 - t_0)))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 0.5 - (Math.cos((1.0 * (phi1 - phi2))) * 0.5);
	return (R * 2.0) * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0)));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = 0.5 - (math.cos((1.0 * (phi1 - phi2))) * 0.5)
	return (R * 2.0) * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0)))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(0.5 - Float64(cos(Float64(1.0 * Float64(phi1 - phi2))) * 0.5))
	return Float64(Float64(R * 2.0) * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = 0.5 - (cos((1.0 * (phi1 - phi2))) * 0.5);
	tmp = (R * 2.0) * atan2(sqrt(t_0), sqrt((1.0 - t_0)));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(0.5 - N[(N[Cos[N[(1.0 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]}, N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 61.8%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

3. Applied rewrites 60.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
4. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

5. Applied rewrites 62.2%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
6. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

7. Applied rewrites 61.9%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
8. Step-by-step derivation

   1. lift-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   2. lift-/.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)}}\right) \]

   3. lift--.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 - \lambda_2}}{2}\right)\right)}}\right) \]

   4. sub-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \left(\frac{\color{blue}{\lambda_1 + \left(\mathsf{neg}\left(\lambda_2\right)\right)}}{2}\right)\right)}}\right) \]

   5. div-add N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \sin \color{blue}{\left(\frac{\lambda_1}{2} + \frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)}}\right) \]

   6. sin-sum N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\left(\sin \left(\frac{\lambda_1}{2}\right) \cdot \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right) + \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\frac{\lambda_1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)}\right)}}\right) \]

   8. lower-sin.f64 N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\color{blue}{\sin \left(\frac{\lambda_1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   9. mult-flip N/A

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   10. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   11. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)}, \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   12. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \color{blue}{\cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   13. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   14. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   15. frac-2neg-rev N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   16. lower-/.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \color{blue}{\left(\frac{\lambda_2}{\mathsf{neg}\left(2\right)}\right)}, \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{\color{blue}{-2}}\right), \cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   19. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \color{blue}{\cos \left(\frac{\lambda_1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   20. mult-flip N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   21. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \color{blue}{\frac{1}{2}}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   22. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \color{blue}{\left(\lambda_1 \cdot \frac{1}{2}\right)} \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)\right)\right)}}\right) \]

   23. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \color{blue}{\sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{2}\right)}\right)\right)}}\right) \]

   24. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\color{blue}{\mathsf{neg}\left(-2\right)}}\right)\right)\right)}}\right) \]

   25. metadata-eval N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\mathsf{neg}\left(\lambda_2\right)}{\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)}\right)\right)\right)}}\right) \]

9. Applied rewrites 76.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)}\right)}}\right) \]

10. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]
11. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    4. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    6. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot \frac{1}{2}\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot \frac{1}{2}\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

    11. lower--.f64 46.4%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

12. Applied rewrites 46.4%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right) \cdot \mathsf{fma}\left(\sin \left(\lambda_1 \cdot 0.5\right), \cos \left(\frac{\lambda_2}{-2}\right), \cos \left(\lambda_1 \cdot 0.5\right) \cdot \sin \left(\frac{\lambda_2}{-2}\right)\right)\right)}}\right) \]

13. Taylor expanded in lambda1 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}\right) + {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]
14. Step-by-step derivation

    1. lower-fma.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \color{blue}{\cos \phi_2} \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower-cos.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\color{blue}{\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    5. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{\color{blue}{2}}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    6. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    7. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    8. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    9. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    10. lower-*.f64 N/A

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    11. lower--.f64 45.9%

        \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

15. Applied rewrites 45.9%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \color{blue}{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}}\right) \]

16. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]
17. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\frac{-1}{2} \cdot \lambda_2\right)}^{2}, {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

    4. lower--.f64 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

18. Applied rewrites 29.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\right) \]

19. Taylor expanded in lambda2 around 0

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
20. Step-by-step derivation

    1. lower-pow.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    2. lower-sin.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    3. lower-*.f64 N/A

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(\frac{1}{2} \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

    4. lower--.f64 29.3%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right) \]

21. Applied rewrites 29.3%

    \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{\color{blue}{2}}}}\right) \]
22. Step-by-step derivation

23. Applied rewrites 26.3%

    \[\leadsto \color{blue}{\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5}}{\sqrt{1 - \left(0.5 - \cos \left(1 \cdot \left(\phi_1 - \phi_2\right)\right) \cdot 0.5\right)}}} \]
24. Add Preprocessing

Reproduce

herbie shell --seed 2025193 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))