Distance on a great circle

Percentage Accurate: 62.5% → 63.9%

Time: 30.5s

Alternatives: 18

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.9% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 64.7

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

4. Applied rewrites 64.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

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

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

6. Applied rewrites 64.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. sqr-cos-a N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

8. Applied rewrites 64.8%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 65.2

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

10. Applied rewrites 65.2%

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

Alternative 2: 63.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 64.7

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

4. Applied rewrites 64.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

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

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

6. Applied rewrites 64.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. sqr-cos-a N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

8. Applied rewrites 64.8%

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

9. Taylor expanded in phi1 around inf

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

    1. *-commutative N/A

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

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

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

    3. *-commutative N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-sin.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. cos-neg-rev N/A

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

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

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

    9. metadata-eval N/A

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

    10. lower-cos.f64 N/A

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

    11. lower-*.f64 N/A

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

    12. sin-neg-rev N/A

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

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

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

    14. metadata-eval N/A

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

    15. cos-neg-rev N/A

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

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

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

    17. metadata-eval N/A

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

    18. lower-*.f64 N/A

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

11. Applied rewrites 64.8%

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

Alternative 3: 41.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-17} \lor \neg \left(t\_0 \leq 10^{-21}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_1 \cdot \cos \phi_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{\mathsf{fma}\left({t\_0}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, {\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0)))
   (if (or (<= t_0 -2e-17) (not (<= t_0 1e-21)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma t_1 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0)))
        (sqrt (- (pow (cos (* -0.5 phi1)) 2.0) (* t_1 (cos phi1)))))))
     (*
      (atan2
       (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0))
       (sqrt
        (fma
         (* (pow t_0 2.0) (- (cos phi2)))
         (cos phi1)
         (pow (cos (/ (- phi1 phi2) 2.0)) 2.0))))
      (* R 2.0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((lambda2 - lambda1) * -0.5)), 2.0);
	double tmp;
	if ((t_0 <= -2e-17) || !(t_0 <= 1e-21)) {
		tmp = R * (2.0 * atan2(sqrt(fma(t_1, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))), sqrt((pow(cos((-0.5 * phi1)), 2.0) - (t_1 * cos(phi1))))));
	} else {
		tmp = atan2(sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0)), sqrt(fma((pow(t_0, 2.0) * -cos(phi2)), cos(phi1), pow(cos(((phi1 - phi2) / 2.0)), 2.0)))) * (R * 2.0);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0
	tmp = 0.0
	if ((t_0 <= -2e-17) || !(t_0 <= 1e-21))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_1, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64((cos(Float64(-0.5 * phi1)) ^ 2.0) - Float64(t_1 * cos(phi1)))))));
	else
		tmp = Float64(atan(sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0)), sqrt(fma(Float64((t_0 ^ 2.0) * Float64(-cos(phi2))), cos(phi1), (cos(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)))) * Float64(R * 2.0));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2e-17], N[Not[LessEqual[t$95$0, 1e-21]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(N[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[Power[N[Cos[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[(R * 2.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2.00000000000000014e-17 or 9.99999999999999908e-22 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 57.9%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 48.3%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 33.4%

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

   if -2.00000000000000014e-17 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 9.99999999999999908e-22

   1. Initial program 84.5%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 81.1%

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

   10. Applied rewrites 81.3%

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

4. Final simplification 43.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -2 \cdot 10^{-17} \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 10^{-21}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, {\cos \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}} \cdot \left(R \cdot 2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 64.7

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

4. Applied rewrites 64.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

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

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

6. Applied rewrites 64.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. sqr-cos-a N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

8. Applied rewrites 64.8%

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

10. Applied rewrites 63.7%

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

11. Final simplification 63.7%

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

Alternative 5: 62.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*l* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. lift-cos.f64 N/A

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

   9. cos-mult N/A

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

   10. associate-*l/ N/A

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

   11. lift-pow.f64 N/A

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

   12. unpow2 N/A

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

   13. lift-sin.f64 N/A

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

4. Applied rewrites 64.0%

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

5. Final simplification 64.0%

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

Alternative 6: 62.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 64.7

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

4. Applied rewrites 64.7%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

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

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

6. Applied rewrites 64.8%

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

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-cos.f64 N/A

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

   4. lift-cos.f64 N/A

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

   5. sqr-cos-a N/A

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

   6. associate-+r+ N/A

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

   7. lower-+.f64 N/A

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

8. Applied rewrites 64.8%

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

   1. lift-+.f64 N/A

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

10. Applied rewrites 63.7%

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

Alternative 7: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (fma
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (cos phi1)
          (pow (sin (* 0.5 phi1)) 2.0))))
   (if (or (<= phi1 -1.6e-5) (not (<= phi1 1.3e+35)))
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (* (cos phi1) (cos phi2)) t_0) t_0)))
        (sqrt
         (-
          (pow (cos (* 0.5 phi2)) 2.0)
          (* (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0) (cos phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if ((phi1 <= -1.6e-5) || !(phi1 <= 1.3e+35)) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt((pow(cos((0.5 * phi2)), 2.0) - (pow(sin(((lambda2 - lambda1) * -0.5)), 2.0) * cos(phi2))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if ((phi1 <= -1.6e-5) || !(phi1 <= 1.3e+35))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_0))), sqrt(Float64((cos(Float64(0.5 * phi2)) ^ 2.0) - Float64((sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0) * cos(phi2)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1.6e-5], N[Not[LessEqual[phi1, 1.3e+35]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.59999999999999993e-5 or 1.30000000000000003e35 < phi1

   1. Initial program 53.9%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 30.6

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

   11. Applied rewrites 30.6%

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

   12. Taylor expanded in phi2 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 56.1

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

   14. Applied rewrites 56.1%

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

   if -1.59999999999999993e-5 < phi1 < 1.30000000000000003e35

   1. Initial program 72.8%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 72.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_2}}}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_2}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.00032 \lor \neg \left(\phi_1 \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, t\_2\right)}}{\sqrt{1 - \left(t\_2 + t\_0 \cdot \cos \phi_2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma t_0 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
   (if (or (<= phi1 -0.00032) (not (<= phi1 1.3e+35)))
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi2)
          (* (cos phi1) (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0))
          t_2))
        (sqrt (- 1.0 (+ t_2 (* t_0 (cos phi2)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double tmp;
	if ((phi1 <= -0.00032) || !(phi1 <= 1.3e+35)) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), (cos(phi1) * pow(sin(((lambda1 - lambda2) / 2.0)), 2.0)), t_2)), sqrt((1.0 - (t_2 + (t_0 * cos(phi2)))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(t_0, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	tmp = 0.0
	if ((phi1 <= -0.00032) || !(phi1 <= 1.3e+35))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), Float64(cos(phi1) * (sin(Float64(Float64(lambda1 - lambda2) / 2.0)) ^ 2.0)), t_2)), sqrt(Float64(1.0 - Float64(t_2 + Float64(t_0 * cos(phi2))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi1, -0.00032], N[Not[LessEqual[phi1, 1.3e+35]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$2 + N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.20000000000000026e-4 or 1.30000000000000003e35 < phi1

   1. Initial program 53.9%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 30.6

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

   11. Applied rewrites 30.6%

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

   12. Taylor expanded in phi2 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 56.1

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

   14. Applied rewrites 56.1%

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

   if -3.20000000000000026e-4 < phi1 < 1.30000000000000003e35

   1. Initial program 72.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 72.7

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

   7. Applied rewrites 72.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00032 \lor \neg \left(\phi_1 \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_2\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma t_0 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0))))
   (if (or (<= phi1 -1.6e-5) (not (<= phi1 1.3e+35)))
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi2)
          (* (cos phi1) (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- 1.0 (fma t_0 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if ((phi1 <= -1.6e-5) || !(phi1 <= 1.3e+35)) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), (cos(phi1) * pow(sin(((lambda1 - lambda2) / 2.0)), 2.0)), pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - fma(t_0, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(t_0, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if ((phi1 <= -1.6e-5) || !(phi1 <= 1.3e+35))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), Float64(cos(phi1) * (sin(Float64(Float64(lambda1 - lambda2) / 2.0)) ^ 2.0)), (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_0, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1.6e-5], N[Not[LessEqual[phi1, 1.3e+35]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.59999999999999993e-5 or 1.30000000000000003e35 < phi1

   1. Initial program 53.9%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 29.8%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 30.6

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

   11. Applied rewrites 30.6%

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

   12. Taylor expanded in phi2 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 56.1

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

   14. Applied rewrites 56.1%

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

   if -1.59999999999999993e-5 < phi1 < 1.30000000000000003e35

   1. Initial program 72.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   4. Applied rewrites 72.7%

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

   5. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 72.7

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

   7. Applied rewrites 72.7%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 1.3 \cdot 10^{+35}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 12800\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, 0.5\right) + \cos \left(\frac{\phi_1 - \phi_2}{-2} \cdot 2\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, \cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_0 \cdot \cos \phi_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0)))
   (if (or (<= phi2 -2.3e-7) (not (<= phi2 12800.0)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (cos phi2)
          (pow (sin (* -0.5 phi2)) 2.0)))
        (sqrt
         (+
          (fma
           (* (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0) (- (cos phi2)))
           (cos phi1)
           0.5)
          (* (cos (* (/ (- phi1 phi2) -2.0) 2.0)) 0.5))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          t_0
          (* (cos phi2) (cos phi1))
          (pow (sin (* (- phi2 phi1) -0.5)) 2.0)))
        (sqrt (- (pow (cos (* -0.5 phi1)) 2.0) (* t_0 (cos phi1))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((lambda2 - lambda1) * -0.5)), 2.0);
	double tmp;
	if ((phi2 <= -2.3e-7) || !(phi2 <= 12800.0)) {
		tmp = R * (2.0 * atan2(sqrt(fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi2), pow(sin((-0.5 * phi2)), 2.0))), sqrt((fma((pow(sin(((lambda1 - lambda2) / 2.0)), 2.0) * -cos(phi2)), cos(phi1), 0.5) + (cos((((phi1 - phi2) / -2.0) * 2.0)) * 0.5)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(t_0, (cos(phi2) * cos(phi1)), pow(sin(((phi2 - phi1) * -0.5)), 2.0))), sqrt((pow(cos((-0.5 * phi1)), 2.0) - (t_0 * cos(phi1))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -2.3e-7) || !(phi2 <= 12800.0))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(fma(Float64((sin(Float64(Float64(lambda1 - lambda2) / 2.0)) ^ 2.0) * Float64(-cos(phi2))), cos(phi1), 0.5) + Float64(cos(Float64(Float64(Float64(phi1 - phi2) / -2.0) * 2.0)) * 0.5))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_0, Float64(cos(phi2) * cos(phi1)), (sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(-0.5 * phi1)) ^ 2.0) - Float64(t_0 * cos(phi1)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -2.3e-7], N[Not[LessEqual[phi2, 12800.0]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] / -2.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$0 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.29999999999999995e-7 or 12800 < phi2

   1. Initial program 45.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

   6. Applied rewrites 47.6%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-cos-a N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

   8. Applied rewrites 47.6%

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

   9. Taylor expanded in phi1 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. unpow2 N/A

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

       9. sqr-neg-rev N/A

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

       10. sin-neg-rev N/A

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

       11. sin-neg-rev N/A

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

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

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

       13. metadata-eval N/A

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

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

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

       15. metadata-eval N/A

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

       16. unpow2 N/A

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

       17. lower-pow.f64 N/A

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

       18. lower-sin.f64 N/A

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

       19. lower-*.f64 46.8

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

   11. Applied rewrites 46.8%

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

   if -2.29999999999999995e-7 < phi2 < 12800

   1. Initial program 80.9%

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

   3. Taylor expanded in phi2 around 0

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in lambda1 around inf

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

   8. Applied rewrites 80.9%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 12800\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, 0.5\right) + \cos \left(\frac{\phi_1 - \phi_2}{-2} \cdot 2\right) \cdot 0.5}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -1.16 \cdot 10^{-16} \lor \neg \left(\phi_1 \leq 5.2 \cdot 10^{-21}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, 0.5\right) + \cos \left(\frac{\phi_1 - \phi_2}{-2} \cdot 2\right) \cdot 0.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma t_0 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0))))
   (if (or (<= phi1 -1.16e-16) (not (<= phi1 5.2e-21)))
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma t_0 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0)))
        (sqrt
         (+
          (fma
           (* (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0) (- (cos phi2)))
           (cos phi1)
           0.5)
          (* (cos (* (/ (- phi1 phi2) -2.0) 2.0)) 0.5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if ((phi1 <= -1.16e-16) || !(phi1 <= 5.2e-21)) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(t_0, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))), sqrt((fma((pow(sin(((lambda1 - lambda2) / 2.0)), 2.0) * -cos(phi2)), cos(phi1), 0.5) + (cos((((phi1 - phi2) / -2.0) * 2.0)) * 0.5)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(t_0, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if ((phi1 <= -1.16e-16) || !(phi1 <= 5.2e-21))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_0, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0))), sqrt(Float64(fma(Float64((sin(Float64(Float64(lambda1 - lambda2) / 2.0)) ^ 2.0) * Float64(-cos(phi2))), cos(phi1), 0.5) + Float64(cos(Float64(Float64(Float64(phi1 - phi2) / -2.0) * 2.0)) * 0.5))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -1.16e-16], N[Not[LessEqual[phi1, 5.2e-21]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] / -2.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.1600000000000001e-16 or 5.20000000000000035e-21 < phi1

   1. Initial program 53.8%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 41.6%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 30.7%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 31.4

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

   11. Applied rewrites 31.4%

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

   12. Taylor expanded in phi2 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 55.6

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

   14. Applied rewrites 55.6%

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

   if -1.1600000000000001e-16 < phi1 < 5.20000000000000035e-21

   1. Initial program 74.5%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 74.5

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

   4. Applied rewrites 74.5%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

   6. Applied rewrites 74.5%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-cos-a N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

   8. Applied rewrites 74.6%

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

   9. Taylor expanded in phi1 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. unpow2 N/A

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

       9. sqr-neg-rev N/A

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

       10. sin-neg-rev N/A

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

       11. sin-neg-rev N/A

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

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

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

       13. metadata-eval N/A

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

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

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

       15. metadata-eval N/A

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

       16. unpow2 N/A

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

       17. lower-pow.f64 N/A

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

       18. lower-sin.f64 N/A

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

       19. lower-*.f64 73.0

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

   11. Applied rewrites 73.0%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.16 \cdot 10^{-16} \lor \neg \left(\phi_1 \leq 5.2 \cdot 10^{-21}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, 0.5\right) + \cos \left(\frac{\phi_1 - \phi_2}{-2} \cdot 2\right) \cdot 0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 8.6 \cdot 10^{-24}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, 0.5\right) + \cos \left(\frac{\phi_1 - \phi_2}{-2} \cdot 2\right) \cdot 0.5}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (if (or (<= phi2 -2.3e-7) (not (<= phi2 8.6e-24)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0))
        (sqrt (- 1.0 (fma t_0 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma t_0 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0)))
        (sqrt
         (+
          (fma
           (* (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0) (- (cos phi2)))
           (cos phi1)
           0.5)
          (* (cos (* (/ (- phi1 phi2) -2.0) 2.0)) 0.5)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double tmp;
	if ((phi2 <= -2.3e-7) || !(phi2 <= 8.6e-24)) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0)), sqrt((1.0 - fma(t_0, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0))), sqrt((fma((pow(sin(((lambda1 - lambda2) / 2.0)), 2.0) * -cos(phi2)), cos(phi1), 0.5) + (cos((((phi1 - phi2) / -2.0) * 2.0)) * 0.5)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -2.3e-7) || !(phi2 <= 8.6e-24))
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0)), sqrt(Float64(1.0 - fma(t_0, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_0, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64(fma(Float64((sin(Float64(Float64(lambda1 - lambda2) / 2.0)) ^ 2.0) * Float64(-cos(phi2))), cos(phi1), 0.5) + Float64(cos(Float64(Float64(Float64(phi1 - phi2) / -2.0) * 2.0)) * 0.5))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -2.3e-7], N[Not[LessEqual[phi2, 8.6e-24]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(N[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(N[(N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * (-N[Cos[phi2], $MachinePrecision])), $MachinePrecision] * N[Cos[phi1], $MachinePrecision] + 0.5), $MachinePrecision] + N[(N[Cos[N[(N[(N[(phi1 - phi2), $MachinePrecision] / -2.0), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.29999999999999995e-7 or 8.6000000000000006e-24 < phi2

   1. Initial program 46.1%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 31.9%

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

   9. Taylor expanded in phi1 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 32.7

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

   11. Applied rewrites 32.7%

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

   if -2.29999999999999995e-7 < phi2 < 8.6000000000000006e-24

   1. Initial program 82.2%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 82.2

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

   4. Applied rewrites 82.2%

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

      1. lift--.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. associate--r+ N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-*l* N/A

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

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

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

   6. Applied rewrites 82.3%

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

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-cos.f64 N/A

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

      4. lift-cos.f64 N/A

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

      5. sqr-cos-a N/A

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

      6. associate-+r+ N/A

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

      7. lower-+.f64 N/A

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

   8. Applied rewrites 82.3%

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

   9. Taylor expanded in phi2 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 80.4

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

   11. Applied rewrites 80.4%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 8.6 \cdot 10^{-24}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{\mathsf{fma}\left({\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2} \cdot \left(-\cos \phi_2\right), \cos \phi_1, 0.5\right) + \cos \left(\frac{\phi_1 - \phi_2}{-2} \cdot 2\right) \cdot 0.5}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 8.6 \cdot 10^{-24}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (fma t_0 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0))))
   (if (or (<= phi2 -2.3e-7) (not (<= phi2 8.6e-24)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0))
        (sqrt (- 1.0 (fma t_0 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0)))))))
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if ((phi2 <= -2.3e-7) || !(phi2 <= 8.6e-24)) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0)), sqrt((1.0 - fma(t_0, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = fma(t_0, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if ((phi2 <= -2.3e-7) || !(phi2 <= 8.6e-24))
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0)), sqrt(Float64(1.0 - fma(t_0, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -2.3e-7], N[Not[LessEqual[phi2, 8.6e-24]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(N[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$1], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.29999999999999995e-7 or 8.6000000000000006e-24 < phi2

   1. Initial program 46.1%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 34.7%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 31.9%

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

   9. Taylor expanded in phi1 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 32.7

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

   11. Applied rewrites 32.7%

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

   if -2.29999999999999995e-7 < phi2 < 8.6000000000000006e-24

   1. Initial program 82.2%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 54.6%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 30.5%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 30.5

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

   11. Applied rewrites 30.5%

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

   12. Taylor expanded in phi2 around 0

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

       1. lower-sqrt.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 80.3

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

   14. Applied rewrites 80.3%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 8.6 \cdot 10^{-24}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -2 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_0}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_0}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_1 \cdot \cos \phi_1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (if (<= (sin (/ (- lambda1 lambda2) 2.0)) -2e-17)
     (*
      R
      (*
       2.0
       (atan2
        t_0
        (sqrt (- 1.0 (fma t_1 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        t_0
        (sqrt
         (-
          1.0
          (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* t_1 (cos phi1)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double tmp;
	if (sin(((lambda1 - lambda2) / 2.0)) <= -2e-17) {
		tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - fma(t_1, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * cos(phi1)))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	tmp = 0.0
	if (sin(Float64(Float64(lambda1 - lambda2) / 2.0)) <= -2e-17)
		tmp = Float64(R * Float64(2.0 * atan(t_0, sqrt(Float64(1.0 - fma(t_1, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_0, sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_1 * cos(phi1))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sqrt[N[Power[N[Sin[N[(N[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], -2e-17], N[(R * N[(2.0 * N[ArcTan[t$95$0 / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$0 / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -2.00000000000000014e-17

   1. Initial program 58.2%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 34.4%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 17.9%

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

   9. Taylor expanded in phi1 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 18.1

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

   11. Applied rewrites 18.1%

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

   if -2.00000000000000014e-17 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 68.1%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 42.4%

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

   9. Taylor expanded in phi2 around 0

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 41.0

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

   11. Applied rewrites 41.0%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \color{blue}{{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} \cdot \cos \phi_1}\right)}}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 29.4% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in lambda2 around 0

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

5. Applied rewrites 44.4%

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

6. Taylor expanded in lambda1 around 0

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

8. Applied rewrites 31.2%

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

10. Applied rewrites 31.3%

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

Alternative 16: 29.2% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in lambda2 around 0

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

5. Applied rewrites 44.4%

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

6. Taylor expanded in lambda1 around 0

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

8. Applied rewrites 31.2%

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

9. Taylor expanded in lambda1 around 0

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

    1. associate-*r* N/A

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

    2. lower-fma.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-cos.f64 N/A

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

    6. lower-cos.f64 N/A

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

    7. lower-pow.f64 N/A

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

    8. lower-sin.f64 N/A

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

    9. lower-*.f64 N/A

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

    10. lower-pow.f64 N/A

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

    11. metadata-eval N/A

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

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

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

    13. lower-sin.f64 N/A

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

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

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

    15. metadata-eval N/A

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

    16. lower-*.f64 N/A

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

    17. lower--.f64 31.2

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

11. Applied rewrites 31.2%

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

Alternative 17: 29.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 4.8 \cdot 10^{+64}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_0}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_0}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0)))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (if (or (<= phi2 -2.3e-7) (not (<= phi2 4.8e+64)))
     (*
      R
      (*
       2.0
       (atan2
        t_0
        (sqrt (- 1.0 (fma t_1 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        t_0
        (sqrt (- 1.0 (fma t_1 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double tmp;
	if ((phi2 <= -2.3e-7) || !(phi2 <= 4.8e+64)) {
		tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - fma(t_1, cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - fma(t_1, cos(phi1), pow(sin((0.5 * phi1)), 2.0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -2.3e-7) || !(phi2 <= 4.8e+64))
		tmp = Float64(R * Float64(2.0 * atan(t_0, sqrt(Float64(1.0 - fma(t_1, cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_0, sqrt(Float64(1.0 - fma(t_1, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sqrt[N[Power[N[Sin[N[(N[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -2.3e-7], N[Not[LessEqual[phi2, 4.8e+64]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[t$95$0 / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$0 / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.29999999999999995e-7 or 4.79999999999999999e64 < phi2

   1. Initial program 44.7%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 34.3%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 31.7%

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

   9. Taylor expanded in phi1 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 32.6

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

   11. Applied rewrites 32.6%

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

   if -2.29999999999999995e-7 < phi2 < 4.79999999999999999e64

   1. Initial program 78.8%

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

   3. Taylor expanded in lambda2 around 0

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

   5. Applied rewrites 52.5%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 30.9%

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

   9. Taylor expanded in phi2 around 0

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

       1. lower--.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. lower-pow.f64 N/A

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

       5. lower-sin.f64 N/A

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

       6. lower-*.f64 N/A

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

       7. lower--.f64 N/A

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

       8. lower-cos.f64 N/A

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

       9. lower-pow.f64 N/A

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

       10. lower-sin.f64 N/A

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

       11. lower-*.f64 31.0

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

   11. Applied rewrites 31.0%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.3 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 4.8 \cdot 10^{+64}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 24.9% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 63.6%

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

3. Taylor expanded in lambda2 around 0

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

5. Applied rewrites 44.4%

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

6. Taylor expanded in lambda1 around 0

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

8. Applied rewrites 31.2%

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

9. Taylor expanded in phi2 around 0

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

    1. lower--.f64 N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. lower-pow.f64 N/A

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

    5. lower-sin.f64 N/A

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

    6. lower-*.f64 N/A

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

    7. lower--.f64 N/A

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

    8. lower-cos.f64 N/A

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

    9. lower-pow.f64 N/A

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

    10. lower-sin.f64 N/A

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

    11. lower-*.f64 26.4

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

11. Applied rewrites 26.4%

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

Reproduce

herbie shell --seed 2024359 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Distance on a great circle"
  :precision binary64
  (* R (* 2.0 (atan2 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))) (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0))) (sin (/ (- lambda1 lambda2) 2.0))))))))))