Distance on a great circle

Percentage Accurate: 62.0% → 63.0%

Time: 30.6s

Alternatives: 18

Speedup: 1.1×

• Could not converge on a ground truth

  1. R = 4.4411130483323123e-243
  2. lambda1 = 9.09700543129646e-120
  3. lambda2 = 9.333583972160261e+194
  4. phi1 = 1.6688958383838472e-113
  5. phi2 = -3.5034901408182357e-308

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: 62.0% 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: 63.0% accurate, 0.8× speedup

Math

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

FPCore

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

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 t_1 = cos(phi2) * cos(phi1);
	return (2.0 * R) * atan2(sqrt(fma(t_1, t_0, pow(sin((0.5 * (phi1 - phi2))), 2.0))), sqrt((1.0 - fma(t_1, t_0, pow(fma(sin((0.5 * phi1)), cos((-0.5 * phi2)), (sin((-0.5 * phi2)) * cos((-0.5 * phi1)))), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0
	t_1 = Float64(cos(phi2) * cos(phi1))
	return Float64(Float64(2.0 * R) * atan(sqrt(fma(t_1, t_0, (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, t_0, (fma(sin(Float64(0.5 * phi1)), cos(Float64(-0.5 * phi2)), Float64(sin(Float64(-0.5 * phi2)) * cos(Float64(-0.5 * phi1)))) ^ 2.0))))))
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]}, Block[{t$95$1 = N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]}, N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(t$95$1 * t$95$0 + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * t$95$0 + 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]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.1%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift--.f64 N/A

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

   4. div-sub N/A

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

   5. sin-diff N/A

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

   6. lower--.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sin.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-cos.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-*.f64 N/A

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

   13. lower-cos.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-sin.f64 N/A

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

   16. lower-/.f64 63.3

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

4. Applied rewrites 63.3%

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

5. Taylor expanded in lambda1 around -inf

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

7. Applied rewrites 63.3%

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

Alternative 2: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = cos(Float64(Float64(phi1 - phi2) / -2.0))
	return 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(fma(t_1, t_1, Float64(Float64(Float64(-cos(phi2)) * cos(phi1)) * (t_0 ^ 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[Cos[N[(N[(phi1 - phi2), $MachinePrecision] / -2.0), $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[(t$95$1 * t$95$1 + N[(N[((-N[Cos[phi2], $MachinePrecision]) * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.1%

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

   1. lift--.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. associate--r+ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

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

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

4. Applied rewrites 62.2%

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

5. Final simplification 62.2%

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

Alternative 3: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.088 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-40}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= t_0 -0.088) (not (<= t_0 5e-40)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (cos phi1)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (sqrt
         (-
          (pow (cos (* -0.5 phi1)) 2.0)
          (* (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0) (cos phi1)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0))
        (sqrt
         (-
          1.0
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((t_0 <= -0.088) || !(t_0 <= 5e-40)) {
		tmp = R * (2.0 * atan2(sqrt(fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi1), pow(sin((0.5 * phi1)), 2.0))), sqrt((pow(cos((-0.5 * phi1)), 2.0) - (pow(sin(((lambda2 - lambda1) * -0.5)), 2.0) * cos(phi1))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0)), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((t_0 <= -0.088) || !(t_0 <= 5e-40))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64((cos(Float64(-0.5 * phi1)) ^ 2.0) - Float64((sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0) * cos(phi1)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0)), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_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]}, If[Or[LessEqual[t$95$0, -0.088], N[Not[LessEqual[t$95$0, 5e-40]], $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[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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[(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]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.087999999999999995 or 4.99999999999999965e-40 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 56.5%

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

   3. Taylor expanded in 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 42.8%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 42.7

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

   8. Applied rewrites 42.7%

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

   if -0.087999999999999995 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 4.99999999999999965e-40

   1. Initial program 74.6%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 69.6%

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

   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 68.5%

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

4. Final simplification 50.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -0.088 \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 5 \cdot 10^{-40}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

4. Applied rewrites 62.2%

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

Alternative 5: 62.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.1%

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

3. Taylor expanded in lambda1 around 0

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

5. Applied rewrites 62.1%

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

Alternative 6: 62.6% 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 62.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. 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 62.7%

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

5. Final simplification 62.7%

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

Alternative 7: 62.6% accurate, 1.1× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0))
        (t_1 (/ (- phi1 phi2) 2.0))
        (t_2 (sin t_1)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (fma t_2 t_2 (* t_0 (* (cos phi2) (cos phi1)))))
      (sqrt
       (-
        1.0
        (/
         (fma
          (+ (cos (- phi1 phi2)) (cos (+ phi2 phi1)))
          t_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 = pow(sin(((lambda1 - lambda2) / 2.0)), 2.0);
	double t_1 = (phi1 - phi2) / 2.0;
	double t_2 = sin(t_1);
	return R * (2.0 * atan2(sqrt(fma(t_2, t_2, (t_0 * (cos(phi2) * cos(phi1))))), sqrt((1.0 - (fma((cos((phi1 - phi2)) + cos((phi2 + phi1))), t_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)) ^ 2.0
	t_1 = Float64(Float64(phi1 - phi2) / 2.0)
	t_2 = sin(t_1)
	return Float64(R * Float64(2.0 * atan(sqrt(fma(t_2, t_2, Float64(t_0 * Float64(cos(phi2) * cos(phi1))))), sqrt(Float64(1.0 - Float64(fma(Float64(cos(Float64(phi1 - phi2)) + cos(Float64(phi2 + phi1))), t_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[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$2 = N[Sin[t$95$1], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 * t$95$2 + N[(t$95$0 * N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $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] * t$95$0 + 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 62.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
2. Add Preprocessing
3. 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 62.7%

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

   1. lift-+.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-fma.f64 62.8

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

   5. lift-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. pow2 N/A

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

   11. lower-pow.f64 62.7

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

   12. lift-*.f64 N/A

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

6. Applied rewrites 62.7%

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

7. Final simplification 62.7%

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

Alternative 8: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_1 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_2 := \mathsf{fma}\left(t\_1, \cos \phi_1, t\_0\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_1 \leq -1.18 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_3\right) \cdot t\_3}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - t\_1 \cdot \cos \phi_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, t\_0\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_1 \cdot \cos \phi_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 phi1)) 2.0))
        (t_1 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (t_2 (fma t_1 (cos phi1) t_0))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= phi1 -1.18e-5)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (if (<= phi1 3.7e-13)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
            (* (* (* (cos phi1) (cos phi2)) t_3) t_3)))
          (sqrt (- (pow (cos (* 0.5 phi2)) 2.0) (* t_1 (cos phi2)))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0) (cos phi1) t_0))
          (sqrt (- (pow (cos (* -0.5 phi1)) 2.0) (* t_1 (cos phi1)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * phi1)), 2.0);
	double t_1 = pow(sin(((lambda2 - lambda1) * -0.5)), 2.0);
	double t_2 = fma(t_1, cos(phi1), t_0);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (phi1 <= -1.18e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else if (phi1 <= 3.7e-13) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_3) * t_3))), sqrt((pow(cos((0.5 * phi2)), 2.0) - (t_1 * cos(phi2))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi1), t_0)), sqrt((pow(cos((-0.5 * phi1)), 2.0) - (t_1 * cos(phi1))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * phi1)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0
	t_2 = fma(t_1, cos(phi1), t_0)
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (phi1 <= -1.18e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	elseif (phi1 <= 3.7e-13)
		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_3) * t_3))), sqrt(Float64((cos(Float64(0.5 * phi2)) ^ 2.0) - Float64(t_1 * cos(phi2)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi1), t_0)), sqrt(Float64((cos(Float64(-0.5 * phi1)) ^ 2.0) - Float64(t_1 * cos(phi1)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1.18e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.7e-13], 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$3), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$1 * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $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[phi1], $MachinePrecision] + t$95$0), $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]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.18000000000000005e-5

   1. Initial program 47.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 49.3%

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

   if -1.18000000000000005e-5 < phi1 < 3.69999999999999989e-13

   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 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 82.1%

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

   if 3.69999999999999989e-13 < phi1

   1. Initial program 43.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 44.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

Alternative 9: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_1 := {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}\\ t_2 := \frac{\phi_1 - \phi_2}{2}\\ t_3 := {\sin t\_2}^{2}\\ t_4 := \mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)\\ \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + t\_0 \cdot \cos \phi_2}}{\sqrt{1 - \frac{\mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_2 + \phi_1\right), t\_1, 1 - \cos \left(2 \cdot t\_2\right)\right)}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot t\_1, t\_3\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_0 \cdot \cos \phi_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (t_1 (pow (sin (/ (- lambda1 lambda2) 2.0)) 2.0))
        (t_2 (/ (- phi1 phi2) 2.0))
        (t_3 (pow (sin t_2) 2.0))
        (t_4 (fma t_0 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0))))
   (if (<= phi1 -1.8e-5)
     (* R (* 2.0 (atan2 (sqrt t_4) (sqrt (- 1.0 t_4)))))
     (if (<= phi1 6.8e+25)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_3 (* t_0 (cos phi2))))
          (sqrt
           (-
            1.0
            (/
             (fma
              (+ (cos (- phi1 phi2)) (cos (+ phi2 phi1)))
              t_1
              (- 1.0 (cos (* 2.0 t_2))))
             2.0))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi2) (* (cos phi1) t_1) t_3))
          (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 t_1 = pow(sin(((lambda1 - lambda2) / 2.0)), 2.0);
	double t_2 = (phi1 - phi2) / 2.0;
	double t_3 = pow(sin(t_2), 2.0);
	double t_4 = fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0));
	double tmp;
	if (phi1 <= -1.8e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - t_4))));
	} else if (phi1 <= 6.8e+25) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + (t_0 * cos(phi2)))), sqrt((1.0 - (fma((cos((phi1 - phi2)) + cos((phi2 + phi1))), t_1, (1.0 - cos((2.0 * t_2)))) / 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), (cos(phi1) * t_1), t_3)), 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
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0)) ^ 2.0
	t_2 = Float64(Float64(phi1 - phi2) / 2.0)
	t_3 = sin(t_2) ^ 2.0
	t_4 = fma(t_0, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))
	tmp = 0.0
	if (phi1 <= -1.8e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - t_4)))));
	elseif (phi1 <= 6.8e+25)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(t_0 * cos(phi2)))), sqrt(Float64(1.0 - Float64(fma(Float64(cos(Float64(phi1 - phi2)) + cos(Float64(phi2 + phi1))), t_1, Float64(1.0 - cos(Float64(2.0 * t_2)))) / 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), Float64(cos(phi1) * t_1), t_3)), 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]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[t$95$2], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -1.8e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 6.8e+25], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(t$95$0 * N[Cos[phi2], $MachinePrecision]), $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] * t$95$1 + N[(1.0 - N[Cos[N[(2.0 * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$3), $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 3 regimes

2. if phi1 < -1.80000000000000005e-5

   1. Initial program 47.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 49.3%

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

   if -1.80000000000000005e-5 < phi1 < 6.79999999999999967e25

   1. Initial program 78.4%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 78.5%

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

   5. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   7. Applied rewrites 78.5%

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

   if 6.79999999999999967e25 < phi1

   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 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 46.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-*l* N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

   7. Applied rewrites 46.1%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{elif}\;\phi_1 \leq 6.8 \cdot 10^{+25}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_2}}{\sqrt{1 - \frac{\mathsf{fma}\left(\cos \left(\phi_1 - \phi_2\right) + \cos \left(\phi_2 + \phi_1\right), {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, 1 - \cos \left(2 \cdot \frac{\phi_1 - \phi_2}{2}\right)\right)}{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot {\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}, {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_1}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_1 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -0.6 \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{-9}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, \cos \phi_2, t\_1\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, t\_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_0, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (t_1 (pow (sin (* -0.5 phi2)) 2.0)))
   (if (or (<= phi2 -0.6) (not (<= phi2 7.5e-9)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma t_0 (cos phi2) t_1))
        (sqrt
         (-
          1.0
          (fma (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0) (cos phi2) t_1))))))
     (*
      (* 2.0 R)
      (atan2
       (sqrt
        (fma
         (* (cos phi2) (cos phi1))
         t_0
         (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
       (sqrt (- 1.0 (fma t_0 (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 = pow(sin(((lambda2 - lambda1) * -0.5)), 2.0);
	double t_1 = pow(sin((-0.5 * phi2)), 2.0);
	double tmp;
	if ((phi2 <= -0.6) || !(phi2 <= 7.5e-9)) {
		tmp = R * (2.0 * atan2(sqrt(fma(t_0, cos(phi2), t_1)), sqrt((1.0 - fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi2), t_1)))));
	} else {
		tmp = (2.0 * R) * atan2(sqrt(fma((cos(phi2) * cos(phi1)), t_0, pow(sin((0.5 * (phi1 - phi2))), 2.0))), sqrt((1.0 - fma(t_0, cos(phi1), pow(sin((0.5 * phi1)), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0
	t_1 = sin(Float64(-0.5 * phi2)) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -0.6) || !(phi2 <= 7.5e-9))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_0, cos(phi2), t_1)), sqrt(Float64(1.0 - fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi2), t_1))))));
	else
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), t_0, (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))), sqrt(Float64(1.0 - fma(t_0, 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[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -0.6], N[Not[LessEqual[phi2, 7.5e-9]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + t$95$1), $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[phi2], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.599999999999999978 or 7.49999999999999933e-9 < phi2

   1. Initial program 50.3%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 47.1%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 47.8

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

   8. Applied rewrites 47.8%

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

   9. Taylor expanded in phi1 around 0

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

   11. Applied rewrites 51.4%

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

   if -0.599999999999999978 < phi2 < 7.49999999999999933e-9

   1. Initial program 78.0%

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

      1. lift-sin.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 78.0

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

   4. Applied rewrites 78.0%

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

   5. Taylor expanded in lambda1 around -inf

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

   7. Applied rewrites 78.0%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 78.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.6 \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{-9}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_2 := \mathsf{fma}\left(t\_0, \cos \phi_1, t\_1\right)\\ \mathbf{if}\;\phi_1 \leq -1.18 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2}}{\sqrt{1 - t\_2}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-13}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, t\_0, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_0, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, t\_1\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_0 \cdot \cos \phi_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (t_1 (pow (sin (* 0.5 phi1)) 2.0))
        (t_2 (fma t_0 (cos phi1) t_1)))
   (if (<= phi1 -1.18e-5)
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (if (<= phi1 3.7e-13)
       (*
        (* 2.0 R)
        (atan2
         (sqrt
          (fma
           (* (cos phi2) (cos phi1))
           t_0
           (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))
         (sqrt (- 1.0 (fma t_0 (cos phi2) (pow (sin (* -0.5 phi2)) 2.0))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0) (cos phi1) t_1))
          (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 t_1 = pow(sin((0.5 * phi1)), 2.0);
	double t_2 = fma(t_0, cos(phi1), t_1);
	double tmp;
	if (phi1 <= -1.18e-5) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else if (phi1 <= 3.7e-13) {
		tmp = (2.0 * R) * atan2(sqrt(fma((cos(phi2) * cos(phi1)), t_0, pow(sin((0.5 * (phi1 - phi2))), 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(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi1), t_1)), 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
	t_1 = sin(Float64(0.5 * phi1)) ^ 2.0
	t_2 = fma(t_0, cos(phi1), t_1)
	tmp = 0.0
	if (phi1 <= -1.18e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	elseif (phi1 <= 3.7e-13)
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(fma(Float64(cos(phi2) * cos(phi1)), t_0, (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 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((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi1), t_1)), 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]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]}, If[LessEqual[phi1, -1.18e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.7e-13], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * t$95$0 + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $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], 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[phi1], $MachinePrecision] + t$95$1), $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 3 regimes

2. if phi1 < -1.18000000000000005e-5

   1. Initial program 47.2%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 49.3%

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

   if -1.18000000000000005e-5 < phi1 < 3.69999999999999989e-13

   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

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sin-diff N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-sin.f64 N/A

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

      9. lower-/.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 N/A

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

      13. lower-cos.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-sin.f64 N/A

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

      16. lower-/.f64 82.2

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

   4. Applied rewrites 82.2%

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

   5. Taylor expanded in lambda1 around -inf

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

   7. Applied rewrites 82.2%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 82.0%

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

   if 3.69999999999999989e-13 < phi1

   1. Initial program 43.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 44.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

Alternative 12: 40.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.18 \lor \neg \left(t\_0 \leq 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) \cdot t\_0\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= t_0 -0.18) (not (<= t_0 1e-15)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (sqrt
         (-
          1.0
          (fma
           (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
           (cos phi2)
           (pow (sin (* -0.5 phi2)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0))
        (sqrt
         (-
          1.0
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* (* (* (cos phi1) (cos phi2)) t_0) t_0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((t_0 <= -0.18) || !(t_0 <= 1e-15)) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((lambda2 - lambda1) * -0.5)), 2.0)), sqrt((1.0 - fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0)), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * t_0) * t_0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((t_0 <= -0.18) || !(t_0 <= 1e-15))
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0)), sqrt(Float64(1.0 - fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0)), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * t_0) * t_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]}, If[Or[LessEqual[t$95$0, -0.18], N[Not[LessEqual[t$95$0, 1e-15]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(N[(lambda2 - lambda1), $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[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[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[(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]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.17999999999999999 or 1.0000000000000001e-15 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 57.1%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 43.5

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in phi2 around 0

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

   11. Applied rewrites 32.1%

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

   if -0.17999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 1.0000000000000001e-15

   1. Initial program 71.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 65.6%

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

   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 63.5%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -0.18 \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 40.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}\\ \mathbf{if}\;t\_0 \leq -0.18 \lor \neg \left(t\_0 \leq 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, t\_1\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* (- phi2 phi1) -0.5)) 2.0)))
   (if (or (<= t_0 -0.18) (not (<= t_0 1e-15)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (sqrt
         (-
          1.0
          (fma
           (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
           (cos phi2)
           (pow (sin (* -0.5 phi2)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_1)
        (sqrt
         (-
          1.0
          (fma
           (* (cos phi2) (cos phi1))
           (pow (sin (* -0.5 lambda2)) 2.0)
           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(((phi2 - phi1) * -0.5)), 2.0);
	double tmp;
	if ((t_0 <= -0.18) || !(t_0 <= 1e-15)) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((lambda2 - lambda1) * -0.5)), 2.0)), sqrt((1.0 - fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi2), pow(sin((-0.5 * phi2)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - fma((cos(phi2) * cos(phi1)), pow(sin((-0.5 * lambda2)), 2.0), t_1)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0
	tmp = 0.0
	if ((t_0 <= -0.18) || !(t_0 <= 1e-15))
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0)), sqrt(Float64(1.0 - fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - fma(Float64(cos(phi2) * cos(phi1)), (sin(Float64(-0.5 * lambda2)) ^ 2.0), t_1))))));
	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[(phi2 - phi1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.18], N[Not[LessEqual[t$95$0, 1e-15]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(N[(lambda2 - lambda1), $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[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 - N[(N[(N[Cos[phi2], $MachinePrecision] * N[Cos[phi1], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * lambda2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $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))) < -0.17999999999999999 or 1.0000000000000001e-15 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 57.1%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 43.5

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

   8. Applied rewrites 43.5%

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

   9. Taylor expanded in phi2 around 0

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

   11. Applied rewrites 32.1%

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

   if -0.17999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 1.0000000000000001e-15

   1. Initial program 71.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 65.6%

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

   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 63.5%

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

   9. Taylor expanded in 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. *-lft-identity N/A

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

       11. metadata-eval N/A

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

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

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

       13. lower-pow.f64 N/A

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

   11. Applied rewrites 63.4%

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

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -0.18 \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2 \cdot \cos \phi_1, {\sin \left(-0.5 \cdot \lambda_2\right)}^{2}, {\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 61.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_2 \cdot \cos \phi_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, \cos \phi_2, t\_0\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_2, t\_0\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* -0.5 phi2)) 2.0))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0)))
   (if (or (<= phi1 -2.4e-14) (not (<= phi1 3.5e-13)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma t_1 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0)))
        (sqrt (- (pow (cos (* -0.5 phi1)) 2.0) (* t_2 (cos phi1)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma t_2 (cos phi2) t_0))
        (sqrt (- 1.0 (fma t_1 (cos phi2) t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((-0.5 * phi2)), 2.0);
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = pow(sin(((lambda2 - lambda1) * -0.5)), 2.0);
	double tmp;
	if ((phi1 <= -2.4e-14) || !(phi1 <= 3.5e-13)) {
		tmp = R * (2.0 * atan2(sqrt(fma(t_1, cos(phi1), pow(sin((0.5 * phi1)), 2.0))), sqrt((pow(cos((-0.5 * phi1)), 2.0) - (t_2 * cos(phi1))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(t_2, cos(phi2), t_0)), sqrt((1.0 - fma(t_1, cos(phi2), t_0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * phi2)) ^ 2.0
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0
	tmp = 0.0
	if ((phi1 <= -2.4e-14) || !(phi1 <= 3.5e-13))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_1, cos(phi1), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64((cos(Float64(-0.5 * phi1)) ^ 2.0) - Float64(t_2 * cos(phi1)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_2, cos(phi2), t_0)), sqrt(Float64(1.0 - fma(t_1, cos(phi2), t_0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi1, -2.4e-14], N[Not[LessEqual[phi1, 3.5e-13]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 * N[Cos[phi1], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $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$2 * N[Cos[phi1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 * N[Cos[phi2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.4e-14 or 3.5000000000000002e-13 < phi1

   1. Initial program 45.6%

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

   3. Taylor expanded in 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 46.5%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 46.7

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

   8. Applied rewrites 46.7%

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

   if -2.4e-14 < phi1 < 3.5000000000000002e-13

   1. Initial program 83.0%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in phi1 around 0

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

   11. Applied rewrites 81.2%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-14} \lor \neg \left(\phi_1 \leq 3.5 \cdot 10^{-13}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2} \cdot \cos \phi_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left({\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_2 := {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\\ t_3 := \mathsf{fma}\left(t\_0, \cos \phi_1, t\_1\right)\\ t_4 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3}}{\sqrt{1 - t\_3}}\right)\\ \mathbf{elif}\;\phi_1 \leq 3.5 \cdot 10^{-13}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_0, \cos \phi_2, t\_2\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_4, \cos \phi_2, t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_4, \cos \phi_1, t\_1\right)}}{\sqrt{{\cos \left(-0.5 \cdot \phi_1\right)}^{2} - t\_0 \cdot \cos \phi_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0))
        (t_1 (pow (sin (* 0.5 phi1)) 2.0))
        (t_2 (pow (sin (* -0.5 phi2)) 2.0))
        (t_3 (fma t_0 (cos phi1) t_1))
        (t_4 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (if (<= phi1 -2.4e-14)
     (* R (* 2.0 (atan2 (sqrt t_3) (sqrt (- 1.0 t_3)))))
     (if (<= phi1 3.5e-13)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma t_0 (cos phi2) t_2))
          (sqrt (- 1.0 (fma t_4 (cos phi2) t_2))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma t_4 (cos phi1) t_1))
          (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 t_1 = pow(sin((0.5 * phi1)), 2.0);
	double t_2 = pow(sin((-0.5 * phi2)), 2.0);
	double t_3 = fma(t_0, cos(phi1), t_1);
	double t_4 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double tmp;
	if (phi1 <= -2.4e-14) {
		tmp = R * (2.0 * atan2(sqrt(t_3), sqrt((1.0 - t_3))));
	} else if (phi1 <= 3.5e-13) {
		tmp = R * (2.0 * atan2(sqrt(fma(t_0, cos(phi2), t_2)), sqrt((1.0 - fma(t_4, cos(phi2), t_2)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(t_4, cos(phi1), t_1)), 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
	t_1 = sin(Float64(0.5 * phi1)) ^ 2.0
	t_2 = sin(Float64(-0.5 * phi2)) ^ 2.0
	t_3 = fma(t_0, cos(phi1), t_1)
	t_4 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	tmp = 0.0
	if (phi1 <= -2.4e-14)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_3), sqrt(Float64(1.0 - t_3)))));
	elseif (phi1 <= 3.5e-13)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_0, cos(phi2), t_2)), sqrt(Float64(1.0 - fma(t_4, cos(phi2), t_2))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(t_4, cos(phi1), t_1)), 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]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * N[Cos[phi1], $MachinePrecision] + t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[phi1, -2.4e-14], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$3], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$3), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.5e-13], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 * N[Cos[phi2], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$4 * N[Cos[phi2], $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$4 * N[Cos[phi1], $MachinePrecision] + t$95$1), $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 3 regimes

2. if phi1 < -2.4e-14

   1. Initial program 47.9%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   5. Applied rewrites 47.5%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

   8. Applied rewrites 48.8%

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

   if -2.4e-14 < phi1 < 3.5000000000000002e-13

   1. Initial program 83.0%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 81.6

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

   8. Applied rewrites 81.6%

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

   9. Taylor expanded in phi1 around 0

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

   11. Applied rewrites 81.2%

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

   if 3.5000000000000002e-13 < phi1

   1. Initial program 43.3%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 44.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 44.5

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

   8. Applied rewrites 44.5%

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

Alternative 16: 35.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}\\ \mathbf{if}\;t\_0 \leq -0.18 \lor \neg \left(t\_0 \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}{t\_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{t\_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (-
           1.0
           (fma
            (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
            (cos phi2)
            (pow (sin (* -0.5 phi2)) 2.0))))))
   (if (or (<= t_0 -0.18) (not (<= t_0 2e-28)))
     (*
      R
      (* 2.0 (atan2 (sqrt (pow (sin (* (- lambda2 lambda1) -0.5)) 2.0)) t_1)))
     (* R (* 2.0 (atan2 (sqrt (pow (sin (* (- phi2 phi1) -0.5)) 2.0)) 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 = sqrt((1.0 - fma(pow(sin((0.5 * (lambda1 - lambda2))), 2.0), cos(phi2), pow(sin((-0.5 * phi2)), 2.0))));
	double tmp;
	if ((t_0 <= -0.18) || !(t_0 <= 2e-28)) {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((lambda2 - lambda1) * -0.5)), 2.0)), t_1));
	} else {
		tmp = R * (2.0 * atan2(sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0)), t_1));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sqrt(Float64(1.0 - fma((sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), cos(phi2), (sin(Float64(-0.5 * phi2)) ^ 2.0))))
	tmp = 0.0
	if ((t_0 <= -0.18) || !(t_0 <= 2e-28))
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(lambda2 - lambda1) * -0.5)) ^ 2.0)), t_1)));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt((sin(Float64(Float64(phi2 - phi1) * -0.5)) ^ 2.0)), t_1)));
	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[Sqrt[N[(1.0 - 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]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.18], N[Not[LessEqual[t$95$0, 2e-28]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[Power[N[Sin[N[(N[(lambda2 - lambda1), $MachinePrecision] * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $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] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < -0.17999999999999999 or 1.99999999999999994e-28 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64)))

   1. Initial program 57.2%

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

   3. Taylor expanded in phi1 around 0

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

   5. Applied rewrites 42.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-pow.f64 N/A

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

      4. lower-sin.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower-sin.f64 N/A

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

      10. lower-*.f64 43.3

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

   8. Applied rewrites 43.3%

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

   9. Taylor expanded in phi2 around 0

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

   11. Applied rewrites 32.1%

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

   if -0.17999999999999999 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) #s(literal 2 binary64))) < 1.99999999999999994e-28

   1. Initial program 71.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 65.8%

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

   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 64.7%

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

   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 47.5

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

   11. Applied rewrites 47.5%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -0.18 \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 2 \cdot 10^{-28}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\lambda_2 - \lambda_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 29.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -56000 \lor \neg \left(\phi_1 \leq 6.8 \cdot 10^{+25}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_0}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_0}{\sqrt{1 - \mathsf{fma}\left(t\_1, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

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 (<= phi1 -56000.0) (not (<= phi1 6.8e+25)))
     (*
      R
      (*
       2.0
       (atan2
        t_0
        (sqrt (- 1.0 (fma t_1 (cos phi1) (pow (sin (* 0.5 phi1)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        t_0
        (sqrt (- 1.0 (fma t_1 (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 = sqrt(pow(sin(((phi2 - phi1) * -0.5)), 2.0));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double tmp;
	if ((phi1 <= -56000.0) || !(phi1 <= 6.8e+25)) {
		tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - fma(t_1, cos(phi1), pow(sin((0.5 * phi1)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - fma(t_1, cos(phi2), pow(sin((-0.5 * phi2)), 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 ((phi1 <= -56000.0) || !(phi1 <= 6.8e+25))
		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)))))));
	else
		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)))))));
	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[phi1, -56000.0], N[Not[LessEqual[phi1, 6.8e+25]], $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], 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]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -56000 or 6.79999999999999967e25 < phi1

   1. Initial program 45.8%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 37.6%

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

   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 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 phi2 around 0

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

       1. *-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-pow.f64 N/A

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

       4. lower-sin.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. lower-cos.f64 N/A

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

       8. lower-pow.f64 N/A

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

       9. lower-sin.f64 N/A

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

       10. lower-*.f64 32.9

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

   11. Applied rewrites 32.9%

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

   if -56000 < phi1 < 6.79999999999999967e25

   1. Initial program 78.1%

      \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
   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 51.8%

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

   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 35.2%

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

   9. Taylor expanded in 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 35.2

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

   11. Applied rewrites 35.2%

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

4. Final simplification 34.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -56000 \lor \neg \left(\phi_1 \leq 6.8 \cdot 10^{+25}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_1, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_2 - \phi_1\right) \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left({\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, \cos \phi_2, {\sin \left(-0.5 \cdot \phi_2\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 25.1% 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 62.1%

   \[R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]
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 44.8%

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

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 33.4%

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

9. Taylor expanded in phi2 around 0

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

    1. *-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. lower-pow.f64 N/A

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

    4. lower-sin.f64 N/A

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

    5. lower-*.f64 N/A

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

    6. lower--.f64 N/A

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

    7. lower-cos.f64 N/A

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

    8. lower-pow.f64 N/A

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

    9. lower-sin.f64 N/A

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

    10. lower-*.f64 27.5

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

11. Applied rewrites 27.5%

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

Reproduce

herbie shell --seed 2024354 
(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))))))))))