Distance on a great circle

Percentage Accurate: 62.3% → 79.5%

Time: 1.2min

Alternatives: 21

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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: 62.3% 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

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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: 79.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ phi1 2.0)))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (cos (/ phi2 2.0)))
        (t_3 (cos (/ phi1 2.0)))
        (t_4 (sin (/ phi2 2.0)))
        (t_5 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    (atan2
     (sqrt (fma t_1 (* t_5 t_5) (pow (- (* t_0 t_2) (* t_3 t_4)) 2.0)))
     (sqrt
      (-
       1.0
       (fma
        t_1
        (-
         0.5
         (*
          0.5
          (+ (* (cos lambda1) (cos lambda2)) (* (sin lambda1) (sin lambda2)))))
        (pow (fma t_0 t_2 (* t_4 (- t_3))) 2.0)))))
    (* 2.0 R))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi1 / 2.0));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = cos((phi2 / 2.0));
	double t_3 = cos((phi1 / 2.0));
	double t_4 = sin((phi2 / 2.0));
	double t_5 = sin(((lambda1 - lambda2) / 2.0));
	return atan2(sqrt(fma(t_1, (t_5 * t_5), pow(((t_0 * t_2) - (t_3 * t_4)), 2.0))), sqrt((1.0 - fma(t_1, (0.5 - (0.5 * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2))))), pow(fma(t_0, t_2, (t_4 * -t_3)), 2.0))))) * (2.0 * R);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi1 / 2.0))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = cos(Float64(phi2 / 2.0))
	t_3 = cos(Float64(phi1 / 2.0))
	t_4 = sin(Float64(phi2 / 2.0))
	t_5 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	return Float64(atan(sqrt(fma(t_1, Float64(t_5 * t_5), (Float64(Float64(t_0 * t_2) - Float64(t_3 * t_4)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, Float64(0.5 - Float64(0.5 * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))), (fma(t_0, t_2, Float64(t_4 * Float64(-t_3))) ^ 2.0))))) * Float64(2.0 * R))
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

    1. cos-diff 79.2%

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

14. Applied egg-rr 79.2%

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

15. Final simplification 79.2%

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

Alternative 2: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{\phi_2}{2}\right)\\ t_1 := \cos \left(\frac{\phi_1}{2}\right)\\ t_2 := \sin \left(\frac{\phi_2}{2}\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \sin \left(\frac{\phi_1}{2}\right)\\ \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \left(e^{\mathsf{log1p}\left(\cos \phi_1\right)} + -1\right), t\_3 \cdot t\_3, {\left(t\_4 \cdot t\_0 - t\_1 \cdot t\_2\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(t\_4, t\_0, t\_2 \cdot \left(-t\_1\right)\right)\right)}^{2}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (/ phi2 2.0)))
        (t_1 (cos (/ phi1 2.0)))
        (t_2 (sin (/ phi2 2.0)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (sin (/ phi1 2.0))))
   (*
    (* 2.0 R)
    (atan2
     (sqrt
      (fma
       (* (cos phi2) (+ (exp (log1p (cos phi1))) -1.0))
       (* t_3 t_3)
       (pow (- (* t_4 t_0) (* t_1 t_2)) 2.0)))
     (sqrt
      (-
       1.0
       (fma
        (* (cos phi1) (cos phi2))
        (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
        (pow (fma t_4 t_0 (* t_2 (- t_1))) 2.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 / 2.0));
	double t_1 = cos((phi1 / 2.0));
	double t_2 = sin((phi2 / 2.0));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = sin((phi1 / 2.0));
	return (2.0 * R) * atan2(sqrt(fma((cos(phi2) * (exp(log1p(cos(phi1))) + -1.0)), (t_3 * t_3), pow(((t_4 * t_0) - (t_1 * t_2)), 2.0))), sqrt((1.0 - fma((cos(phi1) * cos(phi2)), (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(fma(t_4, t_0, (t_2 * -t_1)), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 / 2.0))
	t_1 = cos(Float64(phi1 / 2.0))
	t_2 = sin(Float64(phi2 / 2.0))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = sin(Float64(phi1 / 2.0))
	return Float64(Float64(2.0 * R) * atan(sqrt(fma(Float64(cos(phi2) * Float64(exp(log1p(cos(phi1))) + -1.0)), Float64(t_3 * t_3), (Float64(Float64(t_4 * t_0) - Float64(t_1 * t_2)) ^ 2.0))), sqrt(Float64(1.0 - fma(Float64(cos(phi1) * cos(phi2)), Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (fma(t_4, t_0, Float64(t_2 * Float64(-t_1))) ^ 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

    1. expm1-log1p-u 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos \phi_1\right)\right)} \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    2. expm1-undefine 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \phi_1\right)} - 1\right)} \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

14. Applied egg-rr 78.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\cos \phi_1\right)} - 1\right)} \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

15. Final simplification 78.6%

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2 \cdot \left(e^{\mathsf{log1p}\left(\cos \phi_1\right)} + -1\right), \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(-\cos \left(\frac{\phi_1}{2}\right)\right)\right)\right)}^{2}\right)}} \]
16. Add Preprocessing

Alternative 3: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\phi_2}{2}\right)\\ t_1 := \cos \left(\frac{\phi_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \cos \left(\frac{\phi_1}{2}\right)\\ \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_2, t\_3 \cdot t\_3, {\left(t\_1 \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right)\right)\right) - t\_4 \cdot t\_0\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), t\_1, t\_0 \cdot \left(-t\_4\right)\right)\right)}^{2}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ phi2 2.0)))
        (t_1 (cos (/ phi2 2.0)))
        (t_2 (* (cos phi1) (cos phi2)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (cos (/ phi1 2.0))))
   (*
    (* 2.0 R)
    (atan2
     (sqrt
      (fma
       t_2
       (* t_3 t_3)
       (pow (- (* t_1 (expm1 (log1p (sin (* phi1 0.5))))) (* t_4 t_0)) 2.0)))
     (sqrt
      (-
       1.0
       (fma
        t_2
        (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
        (pow (fma (sin (/ phi1 2.0)) t_1 (* t_0 (- t_4))) 2.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 / 2.0));
	double t_1 = cos((phi2 / 2.0));
	double t_2 = cos(phi1) * cos(phi2);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = cos((phi1 / 2.0));
	return (2.0 * R) * atan2(sqrt(fma(t_2, (t_3 * t_3), pow(((t_1 * expm1(log1p(sin((phi1 * 0.5))))) - (t_4 * t_0)), 2.0))), sqrt((1.0 - fma(t_2, (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(fma(sin((phi1 / 2.0)), t_1, (t_0 * -t_4)), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 / 2.0))
	t_1 = cos(Float64(phi2 / 2.0))
	t_2 = Float64(cos(phi1) * cos(phi2))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = cos(Float64(phi1 / 2.0))
	return Float64(Float64(2.0 * R) * atan(sqrt(fma(t_2, Float64(t_3 * t_3), (Float64(Float64(t_1 * expm1(log1p(sin(Float64(phi1 * 0.5))))) - Float64(t_4 * t_0)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_2, Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (fma(sin(Float64(phi1 / 2.0)), t_1, Float64(t_0 * Float64(-t_4))) ^ 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

    1. expm1-log1p-u 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\phi_1}{2}\right)\right)\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    2. expm1-undefine 78.0%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\frac{\phi_1}{2}\right)\right)} - 1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    3. div-inv 78.0%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\left(e^{\mathsf{log1p}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}\right)} - 1\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    4. metadata-eval 78.0%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\left(e^{\mathsf{log1p}\left(\sin \left(\phi_1 \cdot \color{blue}{0.5}\right)\right)} - 1\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

14. Applied egg-rr 78.0%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right)\right)} - 1\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
15. Step-by-step derivation

    1. expm1-define 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right)\right)\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

16. Simplified 78.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right)\right)\right)} \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

17. Final simplification 78.6%

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\cos \left(\frac{\phi_2}{2}\right) \cdot \mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right)\right)\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(-\cos \left(\frac{\phi_1}{2}\right)\right)\right)\right)}^{2}\right)}} \]
18. Add Preprocessing

Alternative 4: 78.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\phi_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \left(\frac{\phi_1}{2}\right)\\ \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(t\_1, t\_2 \cdot t\_2, {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right) - t\_3 \cdot t\_0\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(t\_1, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), t\_0 \cdot \left(-t\_3\right)\right)\right)}^{2}\right)}} \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ phi2 2.0)))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (cos (/ phi1 2.0))))
   (*
    (* 2.0 R)
    (atan2
     (sqrt
      (fma
       t_1
       (* t_2 t_2)
       (pow
        (-
         (expm1 (log1p (* (sin (* phi1 0.5)) (cos (* phi2 0.5)))))
         (* t_3 t_0))
        2.0)))
     (sqrt
      (-
       1.0
       (fma
        t_1
        (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
        (pow
         (fma (sin (/ phi1 2.0)) (cos (/ phi2 2.0)) (* t_0 (- t_3)))
         2.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 / 2.0));
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = cos((phi1 / 2.0));
	return (2.0 * R) * atan2(sqrt(fma(t_1, (t_2 * t_2), pow((expm1(log1p((sin((phi1 * 0.5)) * cos((phi2 * 0.5))))) - (t_3 * t_0)), 2.0))), sqrt((1.0 - fma(t_1, (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(fma(sin((phi1 / 2.0)), cos((phi2 / 2.0)), (t_0 * -t_3)), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 / 2.0))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = cos(Float64(phi1 / 2.0))
	return Float64(Float64(2.0 * R) * atan(sqrt(fma(t_1, Float64(t_2 * t_2), (Float64(expm1(log1p(Float64(sin(Float64(phi1 * 0.5)) * cos(Float64(phi2 * 0.5))))) - Float64(t_3 * t_0)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (fma(sin(Float64(phi1 / 2.0)), cos(Float64(phi2 / 2.0)), Float64(t_0 * Float64(-t_3))) ^ 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

    1. expm1-log1p-u 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)\right)\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    2. div-inv 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)} \cdot \cos \left(\frac{\phi_2}{2}\right)\right)\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    3. metadata-eval 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot \color{blue}{0.5}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)\right)\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    4. div-inv 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot \frac{1}{2}\right)}\right)\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    5. metadata-eval 78.6%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot \color{blue}{0.5}\right)\right)\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

14. Applied egg-rr 78.6%

    \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)} - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \cos \left(\frac{\phi_1}{2}\right) \cdot \left(-\sin \left(\frac{\phi_2}{2}\right)\right)\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

15. Final simplification 78.6%

    \[\leadsto \left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(-\cos \left(\frac{\phi_1}{2}\right)\right)\right)\right)}^{2}\right)}} \]
16. Add Preprocessing

Alternative 5: 79.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ phi2 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (sin (/ phi1 2.0)))
        (t_3 (cos (/ phi1 2.0)))
        (t_4 (* (cos phi1) (cos phi2)))
        (t_5 (cos (/ phi2 2.0))))
   (*
    (* 2.0 R)
    (atan2
     (sqrt (fma t_4 (* t_1 t_1) (pow (- (* t_2 t_5) (* t_3 t_0)) 2.0)))
     (sqrt
      (-
       1.0
       (fma
        t_4
        (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
        (pow (fma t_2 t_5 (* t_0 (- t_3))) 2.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((phi2 / 2.0));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sin((phi1 / 2.0));
	double t_3 = cos((phi1 / 2.0));
	double t_4 = cos(phi1) * cos(phi2);
	double t_5 = cos((phi2 / 2.0));
	return (2.0 * R) * atan2(sqrt(fma(t_4, (t_1 * t_1), pow(((t_2 * t_5) - (t_3 * t_0)), 2.0))), sqrt((1.0 - fma(t_4, (0.5 - (0.5 * cos((lambda1 - lambda2)))), pow(fma(t_2, t_5, (t_0 * -t_3)), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 / 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(phi1 / 2.0))
	t_3 = cos(Float64(phi1 / 2.0))
	t_4 = Float64(cos(phi1) * cos(phi2))
	t_5 = cos(Float64(phi2 / 2.0))
	return Float64(Float64(2.0 * R) * atan(sqrt(fma(t_4, Float64(t_1 * t_1), (Float64(Float64(t_2 * t_5) - Float64(t_3 * t_0)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_4, Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), (fma(t_2, t_5, Float64(t_0 * Float64(-t_3))) ^ 2.0))))))
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

13. Final simplification 78.6%

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

Alternative 6: 78.9% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = ((sin((phi1 * 0.5d0)) * cos((phi2 * 0.5d0))) - (cos((phi1 * 0.5d0)) * sin((phi2 * 0.5d0)))) ** 2.0d0
    code = 2.0d0 * (r * atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0))) + t_0)), sqrt((1.0d0 + ((cos(phi1) * (cos(phi2) * ((0.5d0 * cos((lambda1 - lambda2))) - 0.5d0))) - t_0)))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(((math.sin((phi1 * 0.5)) * math.cos((phi2 * 0.5))) - (math.cos((phi1 * 0.5)) * math.sin((phi2 * 0.5)))), 2.0)
	return 2.0 * (R * math.atan2(math.sqrt(((math.cos(phi1) * (math.cos(phi2) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0))) + t_0)), math.sqrt((1.0 + ((math.cos(phi1) * (math.cos(phi2) * ((0.5 * math.cos((lambda1 - lambda2))) - 0.5))) - t_0)))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = ((sin((phi1 * 0.5)) * cos((phi2 * 0.5))) - (cos((phi1 * 0.5)) * sin((phi2 * 0.5)))) ^ 2.0;
	tmp = 2.0 * (R * atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0))) + t_0)), sqrt((1.0 + ((cos(phi1) * (cos(phi2) * ((0.5 * cos((lambda1 - lambda2))) - 0.5))) - t_0)))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

13. Taylor expanded in phi1 around 0 78.6%

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

14. Final simplification 78.6%

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

Alternative 7: 63.3% accurate, 0.8× speedup

Math

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

FPCore

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

Julia

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

8. Applied egg-rr 78.6%

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

   1. fmm-def 78.6%

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

   2. distribute-rgt-neg-in 78.6%

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

10. Simplified 78.6%

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

    1. sqr-sin-a 78.6%

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

    2. cos-2 78.6%

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

    3. cos-sum 78.6%

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

    4. add-log-exp 24.5%

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

    5. add-log-exp 24.5%

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

    6. sum-log 24.5%

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

    7. exp-sqrt 24.5%

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

    8. exp-sqrt 24.5%

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

    9. add-sqr-sqrt 24.5%

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

    10. add-log-exp 78.6%

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

12. Applied egg-rr 78.6%

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

    1. distribute-rgt-neg-out 78.6%

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

    2. fmm-def 78.6%

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

    3. sin-diff 63.6%

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

    4. div-sub 63.6%

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

    5. unpow2 63.6%

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

    6. sin-mult 63.7%

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

14. Applied egg-rr 63.7%

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

    1. div-sub 63.7%

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

    2. +-inverses 63.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), \frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(0.5 \cdot \left(\phi_1 - \phi_2\right) + 0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}{2}\right)}} \cdot \left(R \cdot 2\right) \]

    3. +-inverses 63.7%

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

    4. +-inverses 63.7%

       \[\leadsto \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right), {\left(\sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right) - \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), \frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(0.5 \cdot \left(\phi_1 - \phi_2\right) + 0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}{2}\right)}} \cdot \left(R \cdot 2\right) \]

    5. cos-0 63.7%

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

    6. metadata-eval 63.7%

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

    7. distribute-rgt-out 63.7%

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

    8. metadata-eval 63.7%

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

    9. *-rgt-identity 63.7%

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

16. Simplified 63.7%

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

17. Final simplification 63.7%

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

Alternative 8: 63.3% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = cos(phi1) * (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0))
    code = (2.0d0 * r) * atan2(sqrt((t_0 + (((sin((phi1 * 0.5d0)) * cos((phi2 * 0.5d0))) - (cos((phi1 * 0.5d0)) * sin((phi2 * 0.5d0)))) ** 2.0d0))), sqrt((1.0d0 - (t_0 + (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * (math.cos(phi2) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0))
	return (2.0 * R) * math.atan2(math.sqrt((t_0 + math.pow(((math.sin((phi1 * 0.5)) * math.cos((phi2 * 0.5))) - (math.cos((phi1 * 0.5)) * math.sin((phi2 * 0.5)))), 2.0))), math.sqrt((1.0 - (t_0 + math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0));
	tmp = (2.0 * R) * atan2(sqrt((t_0 + (((sin((phi1 * 0.5)) * cos((phi2 * 0.5))) - (cos((phi1 * 0.5)) * sin((phi2 * 0.5)))) ^ 2.0))), sqrt((1.0 - (t_0 + (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*r* 62.6%

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

   2. *-commutative 62.6%

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

3. Simplified 62.6%

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

   1. div-sub 62.6%

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

   2. sin-diff 63.6%

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

6. Applied egg-rr 63.6%

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

7. Taylor expanded in phi1 around 0 63.6%

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

8. Final simplification 63.6%

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

Alternative 9: 62.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.6%

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

   1. associate-*l* 62.6%

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

3. Simplified 62.5%

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

6. Applied egg-rr 63.1%

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

   1. unpow1/2 63.1%

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

   2. unpow2 63.1%

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

   3. rem-sqrt-square 63.1%

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

   4. cancel-sign-sub-inv 63.1%

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

   5. metadata-eval 63.1%

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

   6. *-commutative 63.1%

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

8. Simplified 63.1%

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

9. Final simplification 63.1%

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

Alternative 10: 63.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_2 := \cos \phi_1 \cdot t\_1\\ \mathbf{if}\;\phi_1 \leq -3.9 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 0.00025\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - t\_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(t\_0 \cdot t\_0\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot t\_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_2 (* (cos phi1) t_1)))
   (if (or (<= phi1 -3.9e-5) (not (<= phi1 0.00025)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (pow (sin (* phi1 0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi1 0.5)) 2.0) t_2)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* (* (cos phi1) (cos phi2)) (* t_0 t_0))))
        (sqrt
         (- 1.0 (+ (* (cos phi2) t_1) (pow (sin (* phi2 -0.5)) 2.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = cos(phi1) * t_1;
	double tmp;
	if ((phi1 <= -3.9e-5) || !(phi1 <= 0.00025)) {
		tmp = R * (2.0 * atan2(sqrt((t_2 + pow(sin((phi1 * 0.5)), 2.0))), sqrt((pow(cos((phi1 * 0.5)), 2.0) - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + ((cos(phi1) * cos(phi2)) * (t_0 * t_0)))), sqrt((1.0 - ((cos(phi2) * t_1) + pow(sin((phi2 * -0.5)), 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    t_2 = cos(phi1) * t_1
    if ((phi1 <= (-3.9d-5)) .or. (.not. (phi1 <= 0.00025d0))) then
        tmp = r * (2.0d0 * atan2(sqrt((t_2 + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt(((cos((phi1 * 0.5d0)) ** 2.0d0) - t_2))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + ((cos(phi1) * cos(phi2)) * (t_0 * t_0)))), sqrt((1.0d0 - ((cos(phi2) * t_1) + (sin((phi2 * (-0.5d0))) ** 2.0d0))))))
    end if
    code = tmp
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(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = Math.cos(phi1) * t_1;
	double tmp;
	if ((phi1 <= -3.9e-5) || !(phi1 <= 0.00025)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_2 + Math.pow(Math.sin((phi1 * 0.5)), 2.0))), Math.sqrt((Math.pow(Math.cos((phi1 * 0.5)), 2.0) - t_2))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((Math.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0)))), Math.sqrt((1.0 - ((Math.cos(phi2) * t_1) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	t_2 = math.cos(phi1) * t_1
	tmp = 0
	if (phi1 <= -3.9e-5) or not (phi1 <= 0.00025):
		tmp = R * (2.0 * math.atan2(math.sqrt((t_2 + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((math.pow(math.cos((phi1 * 0.5)), 2.0) - t_2))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + ((math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)))), math.sqrt((1.0 - ((math.cos(phi2) * t_1) + math.pow(math.sin((phi2 * -0.5)), 2.0))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_2 = Float64(cos(phi1) * t_1)
	tmp = 0.0
	if ((phi1 <= -3.9e-5) || !(phi1 <= 0.00025))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + (sin(Float64(phi1 * 0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi1 * 0.5)) ^ 2.0) - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(cos(phi1) * cos(phi2)) * Float64(t_0 * t_0)))), sqrt(Float64(1.0 - Float64(Float64(cos(phi2) * t_1) + (sin(Float64(phi2 * -0.5)) ^ 2.0)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	t_2 = cos(phi1) * t_1;
	tmp = 0.0;
	if ((phi1 <= -3.9e-5) || ~((phi1 <= 0.00025)))
		tmp = R * (2.0 * atan2(sqrt((t_2 + (sin((phi1 * 0.5)) ^ 2.0))), sqrt(((cos((phi1 * 0.5)) ^ 2.0) - t_2))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + ((cos(phi1) * cos(phi2)) * (t_0 * t_0)))), sqrt((1.0 - ((cos(phi2) * t_1) + (sin((phi2 * -0.5)) ^ 2.0))))));
	end
	tmp_2 = 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[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]}, If[Or[LessEqual[phi1, -3.9e-5], N[Not[LessEqual[phi1, 0.00025]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.8999999999999999e-5 or 2.5000000000000001e-4 < phi1

   1. Initial program 46.9%

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

   3. Taylor expanded in phi1 around inf 47.1%

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

      1. expm1-log1p-u 47.1%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left({\sin \left(\frac{\phi_1}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)\right)}}}\right) \]

      2. associate--r+ 47.1%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(1 - {\sin \left(\frac{\phi_1}{2}\right)}^{2}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right) \]

      3. unpow2 47.1%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      4. 1-sub-sin 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\cos \left(\frac{\phi_1}{2}\right)}^{2}} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. div-inv 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. metadata-eval 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot \color{blue}{0.5}\right)}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*r* 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)\right)}}\right) \]

      9. associate-*l* 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}\right)\right)}}\right) \]

      10. pow2 47.2%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)\right)\right)}}\right) \]

   5. Applied egg-rr 47.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)\right)\right)}}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}\right)\right)}}\right) \]

      2. *-commutative 47.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

   7. Simplified 47.2%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)\right)}}}\right) \]

   8. Taylor expanded in phi2 around 0 48.0%

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

      1. *-commutative 48.0%

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

   10. Simplified 48.0%

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

   11. Taylor expanded in phi2 around 0 48.9%

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

   if -3.8999999999999999e-5 < phi1 < 2.5000000000000001e-4

   1. Initial program 79.2%

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

      1. associate-*l* 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in phi1 around 0 79.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.9 \cdot 10^{-5} \lor \neg \left(\phi_1 \leq 0.00025\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left(\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 62.4% accurate, 1.1× speedup

Math

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

FPCore

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + t_1)), sqrt(((1.0d0 + ((cos((phi1 - phi2)) / 2.0d0) - 0.5d0)) - 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.cos(phi1) * Math.cos(phi2)) * (t_0 * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), Math.sqrt(((1.0 + ((Math.cos((phi1 - phi2)) / 2.0) - 0.5)) - t_1))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = (math.cos(phi1) * math.cos(phi2)) * (t_0 * t_0)
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), math.sqrt(((1.0 + ((math.cos((phi1 - phi2)) / 2.0) - 0.5)) - t_1))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = (cos(phi1) * cos(phi2)) * (t_0 * t_0);
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), sqrt(((1.0 + ((cos((phi1 - phi2)) / 2.0) - 0.5)) - 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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * t$95$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] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 + N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 62.6%

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

   1. associate-*l* 62.6%

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

3. Simplified 62.5%

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

   1. unpow2 62.5%

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

   2. sin-mult 62.6%

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

   3. div-inv 62.6%

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

   4. metadata-eval 62.6%

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

   5. div-inv 62.6%

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

   6. metadata-eval 62.6%

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

   7. div-inv 62.6%

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

   8. metadata-eval 62.6%

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

   9. div-inv 62.6%

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

   10. metadata-eval 62.6%

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

6. Applied egg-rr 62.6%

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

   1. div-sub 62.6%

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

   2. +-inverses 62.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(\left(\phi_1 - \phi_2\right) \cdot 0.5 + \left(\phi_1 - \phi_2\right) \cdot 0.5\right)}{2}\right)\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right) \]

   3. cos-0 62.6%

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

   4. metadata-eval 62.6%

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

   5. distribute-lft-out 62.6%

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

   6. metadata-eval 62.6%

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

   7. *-rgt-identity 62.6%

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

8. Simplified 62.6%

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

9. Final simplification 62.6%

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

Alternative 12: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + t\_0 \cdot \left(t\_1 \cdot t\_1\right)}}{\sqrt{\left(1 - t\_2\right) - t\_0 \cdot \left(-1 + \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot -0.5 + 1.5\right)\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (+ t_2 (* t_0 (* t_1 t_1))))
      (sqrt
       (-
        (- 1.0 t_2)
        (* t_0 (+ -1.0 (+ (* (cos (- lambda1 lambda2)) -0.5) 1.5))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	return R * (2.0 * atan2(sqrt((t_2 + (t_0 * (t_1 * t_1)))), sqrt(((1.0 - t_2) - (t_0 * (-1.0 + ((cos((lambda1 - lambda2)) * -0.5) + 1.5)))))));
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    real(8) :: t_2
    t_0 = cos(phi1) * cos(phi2)
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    code = r * (2.0d0 * atan2(sqrt((t_2 + (t_0 * (t_1 * t_1)))), sqrt(((1.0d0 - t_2) - (t_0 * ((-1.0d0) + ((cos((lambda1 - lambda2)) * (-0.5d0)) + 1.5d0)))))))
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi1) * Math.cos(phi2);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	return R * (2.0 * Math.atan2(Math.sqrt((t_2 + (t_0 * (t_1 * t_1)))), Math.sqrt(((1.0 - t_2) - (t_0 * (-1.0 + ((Math.cos((lambda1 - lambda2)) * -0.5) + 1.5)))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.cos(phi2)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	return R * (2.0 * math.atan2(math.sqrt((t_2 + (t_0 * (t_1 * t_1)))), math.sqrt(((1.0 - t_2) - (t_0 * (-1.0 + ((math.cos((lambda1 - lambda2)) * -0.5) + 1.5)))))))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(t_0 * Float64(t_1 * t_1)))), sqrt(Float64(Float64(1.0 - t_2) - Float64(t_0 * Float64(-1.0 + Float64(Float64(cos(Float64(lambda1 - lambda2)) * -0.5) + 1.5))))))))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * cos(phi2);
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	tmp = R * (2.0 * atan2(sqrt((t_2 + (t_0 * (t_1 * t_1)))), sqrt(((1.0 - t_2) - (t_0 * (-1.0 + ((cos((lambda1 - lambda2)) * -0.5) + 1.5)))))));
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(t$95$0 * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] - N[(t$95$0 * N[(-1.0 + N[(N[(N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] + 1.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 62.6%

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

   1. associate-*l* 62.6%

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

3. Simplified 62.5%

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

   1. expm1-log1p-u 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}}\right) \]

   2. expm1-undefine 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)} - 1\right)}}}\right) \]

6. Applied egg-rr 62.5%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} - 1\right)}}}\right) \]
7. Step-by-step derivation

   1. sub-neg 62.5%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)} + \left(-1\right)\right)}}}\right) \]

   2. log1p-undefine 62.5%

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

   3. rem-exp-log 62.5%

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

   4. sub-neg 62.5%

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

   5. associate-+r+ 62.6%

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

   6. metadata-eval 62.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(\color{blue}{1.5} + \left(-0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right)\right) + \left(-1\right)\right)}}\right) \]

   7. distribute-lft-neg-in 62.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(1.5 + \color{blue}{\left(-0.5\right) \cdot \cos \left(\lambda_1 - \lambda_2\right)}\right) + \left(-1\right)\right)}}\right) \]

   8. metadata-eval 62.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(1.5 + \color{blue}{-0.5} \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \left(-1\right)\right)}}\right) \]

   9. metadata-eval 62.6%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{-1}\right)}}\right) \]

8. Simplified 62.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \color{blue}{\left(\left(1.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right)\right) + -1\right)}}}\right) \]

9. Final simplification 62.6%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(-1 + \left(\cos \left(\lambda_1 - \lambda_2\right) \cdot -0.5 + 1.5\right)\right)}}\right) \]
10. Add Preprocessing

Alternative 13: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot 0.5\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -2.05 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 0.000105\right):\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin t\_0}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_1, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t\_2 \cdot \left(t\_1 \cdot t\_2\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot \left(\frac{\cos \left(2 \cdot t\_0\right)}{2} - 0.5\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- lambda1 lambda2) 0.5))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= phi2 -2.05e-5) (not (<= phi2 0.000105)))
     (*
      (* 2.0 R)
      (atan2
       (sqrt
        (+ (* (cos phi2) (pow (sin t_0) 2.0)) (pow (sin (* phi2 -0.5)) 2.0)))
       (sqrt
        (-
         1.0
         (fma
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          t_1
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* t_2 (* t_1 t_2))))
        (sqrt
         (+
          (pow (cos (* phi1 0.5)) 2.0)
          (* (cos phi1) (- (/ (cos (* 2.0 t_0)) 2.0) 0.5))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (lambda1 - lambda2) * 0.5;
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -2.05e-5) || !(phi2 <= 0.000105)) {
		tmp = (2.0 * R) * atan2(sqrt(((cos(phi2) * pow(sin(t_0), 2.0)) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((1.0 - fma((0.5 - (0.5 * cos((lambda1 - lambda2)))), t_1, pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_2 * (t_1 * t_2)))), sqrt((pow(cos((phi1 * 0.5)), 2.0) + (cos(phi1) * ((cos((2.0 * t_0)) / 2.0) - 0.5))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(lambda1 - lambda2) * 0.5)
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((phi2 <= -2.05e-5) || !(phi2 <= 0.000105))
		tmp = Float64(Float64(2.0 * R) * atan(sqrt(Float64(Float64(cos(phi2) * (sin(t_0) ^ 2.0)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), t_1, (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_2 * Float64(t_1 * t_2)))), sqrt(Float64((cos(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * Float64(Float64(cos(Float64(2.0 * t_0)) / 2.0) - 0.5)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -2.05e-5], N[Not[LessEqual[phi2, 0.000105]], $MachinePrecision]], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[t$95$0], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1 + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $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[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$2 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[(N[Cos[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -2.05000000000000002e-5 or 1.05e-4 < phi2

   1. Initial program 43.8%

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

   3. Simplified 43.8%

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

   6. Applied egg-rr 23.5%

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

      1. unpow1 23.5%

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

   8. Simplified 23.5%

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

   9. Taylor expanded in phi1 around 0 45.2%

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

   if -2.05000000000000002e-5 < phi2 < 1.05e-4

   1. Initial program 83.2%

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

   3. Taylor expanded in phi1 around inf 83.2%

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

      1. expm1-log1p-u 83.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left({\sin \left(\frac{\phi_1}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)\right)}}}\right) \]

      2. associate--r+ 83.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(1 - {\sin \left(\frac{\phi_1}{2}\right)}^{2}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right) \]

      3. unpow2 83.2%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      4. 1-sub-sin 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\cos \left(\frac{\phi_1}{2}\right)}^{2}} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. div-inv 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. metadata-eval 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot \color{blue}{0.5}\right)}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*r* 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)\right)}}\right) \]

      9. associate-*l* 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}\right)\right)}}\right) \]

      10. pow2 83.3%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)\right)\right)}}\right) \]

   5. Applied egg-rr 83.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)\right)\right)}}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}\right)\right)}}\right) \]

      2. *-commutative 83.3%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

   7. Simplified 83.3%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)\right)}}}\right) \]

   8. Taylor expanded in phi2 around 0 83.3%

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

      1. *-commutative 83.3%

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

   10. Simplified 83.3%

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

       1. *-commutative 83.3%

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

       2. metadata-eval 83.3%

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

       3. div-inv 83.3%

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

       4. pow2 83.3%

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

       5. sin-mult 83.4%

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

       6. div-inv 83.4%

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

       7. metadata-eval 83.4%

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

       8. div-inv 83.4%

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

       9. metadata-eval 83.4%

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

       10. cos-sum 83.3%

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

       11. cos-2 83.4%

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

       12. div-inv 83.4%

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

       13. metadata-eval 83.4%

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

   12. Applied egg-rr 83.4%

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

       1. div-sub 83.4%

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

       2. +-inverses 83.4%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}{2}\right)}}\right) \]

       3. cos-0 83.4%

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

       4. metadata-eval 83.4%

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

       5. *-commutative 83.4%

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

   14. Simplified 83.4%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.05 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 0.000105\right):\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot \left(\frac{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}{2} - 0.5\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 53.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ t_1 := \cos \phi_1 \cdot {t\_0}^{2}\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-24} \lor \neg \left(\phi_1 \leq 1.85 \cdot 10^{-50}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_1 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), t\_0 \cdot \sqrt{t\_2}\right)}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_2, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* (- lambda1 lambda2) 0.5)))
        (t_1 (* (cos phi1) (pow t_0 2.0)))
        (t_2 (* (cos phi1) (cos phi2))))
   (if (or (<= phi1 -5.2e-24) (not (<= phi1 1.85e-50)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_1 (pow (sin (* phi1 0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi1 0.5)) 2.0) t_1)))))
     (*
      (* 2.0 R)
      (atan2
       (hypot (sin (* 0.5 (- phi1 phi2))) (* t_0 (sqrt t_2)))
       (sqrt
        (-
         1.0
         (fma
          (- 0.5 (* 0.5 (cos (- lambda1 lambda2))))
          t_2
          (pow (sin (* phi2 -0.5)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) * 0.5));
	double t_1 = cos(phi1) * pow(t_0, 2.0);
	double t_2 = cos(phi1) * cos(phi2);
	double tmp;
	if ((phi1 <= -5.2e-24) || !(phi1 <= 1.85e-50)) {
		tmp = R * (2.0 * atan2(sqrt((t_1 + pow(sin((phi1 * 0.5)), 2.0))), sqrt((pow(cos((phi1 * 0.5)), 2.0) - t_1))));
	} else {
		tmp = (2.0 * R) * atan2(hypot(sin((0.5 * (phi1 - phi2))), (t_0 * sqrt(t_2))), sqrt((1.0 - fma((0.5 - (0.5 * cos((lambda1 - lambda2)))), t_2, pow(sin((phi2 * -0.5)), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) * 0.5))
	t_1 = Float64(cos(phi1) * (t_0 ^ 2.0))
	t_2 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if ((phi1 <= -5.2e-24) || !(phi1 <= 1.85e-50))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + (sin(Float64(phi1 * 0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi1 * 0.5)) ^ 2.0) - t_1)))));
	else
		tmp = Float64(Float64(2.0 * R) * atan(hypot(sin(Float64(0.5 * Float64(phi1 - phi2))), Float64(t_0 * sqrt(t_2))), sqrt(Float64(1.0 - fma(Float64(0.5 - Float64(0.5 * cos(Float64(lambda1 - lambda2)))), t_2, (sin(Float64(phi2 * -0.5)) ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -5.2e-24], N[Not[LessEqual[phi1, 1.85e-50]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * R), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(t$95$0 * N[Sqrt[t$95$2], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(0.5 - N[(0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$2 + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.2e-24 or 1.85e-50 < phi1

   1. Initial program 50.4%

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

   3. Taylor expanded in phi1 around inf 46.7%

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

      1. expm1-log1p-u 46.7%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left({\sin \left(\frac{\phi_1}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)\right)}}}\right) \]

      2. associate--r+ 46.7%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(1 - {\sin \left(\frac{\phi_1}{2}\right)}^{2}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right) \]

      3. unpow2 46.7%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      4. 1-sub-sin 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      5. pow2 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\cos \left(\frac{\phi_1}{2}\right)}^{2}} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      6. div-inv 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      7. metadata-eval 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot \color{blue}{0.5}\right)}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

      8. associate-*r* 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)\right)}}\right) \]

      9. associate-*l* 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}\right)\right)}}\right) \]

      10. pow2 46.8%

          \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)\right)\right)}}\right) \]

   5. Applied egg-rr 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)\right)\right)}}}\right) \]
   6. Step-by-step derivation

      1. associate-*r* 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}\right)\right)}}\right) \]

      2. *-commutative 46.8%

         \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

   7. Simplified 46.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)\right)}}}\right) \]

   8. Taylor expanded in phi2 around 0 47.6%

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

      1. *-commutative 47.6%

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

   10. Simplified 47.6%

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

   11. Taylor expanded in phi2 around 0 48.2%

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

   if -5.2e-24 < phi1 < 1.85e-50

   1. Initial program 79.3%

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

   3. Simplified 79.2%

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

   6. Applied egg-rr 66.5%

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

      1. unpow1 66.5%

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

   8. Simplified 66.5%

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

   9. Taylor expanded in phi1 around 0 66.5%

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

       1. *-commutative 66.5%

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

   11. Simplified 66.5%

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

4. Final simplification 55.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.2 \cdot 10^{-24} \lor \neg \left(\phi_1 \leq 1.85 \cdot 10^{-50}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(2 \cdot R\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(0.5 - 0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), \cos \phi_1 \cdot \cos \phi_2, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 48.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_0 * ((cos(phi1) * cos(phi2)) * t_0)))), sqrt(((cos((phi1 * 0.5d0)) ** 2.0d0) + (cos(phi1) * ((cos((2.0d0 * ((lambda1 - lambda2) * 0.5d0))) / 2.0d0) - 0.5d0))))))
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));
	return R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0)))), Math.sqrt((Math.pow(Math.cos((phi1 * 0.5)), 2.0) + (Math.cos(phi1) * ((Math.cos((2.0 * ((lambda1 - lambda2) * 0.5))) / 2.0) - 0.5))))));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)))), math.sqrt((math.pow(math.cos((phi1 * 0.5)), 2.0) + (math.cos(phi1) * ((math.cos((2.0 * ((lambda1 - lambda2) * 0.5))) / 2.0) - 0.5))))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_0 * ((cos(phi1) * cos(phi2)) * t_0)))), sqrt(((cos((phi1 * 0.5)) ^ 2.0) + (cos(phi1) * ((cos((2.0 * ((lambda1 - lambda2) * 0.5))) / 2.0) - 0.5))))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

   1. expm1-log1p-u 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left({\sin \left(\frac{\phi_1}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)\right)}}}\right) \]

   2. associate--r+ 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(1 - {\sin \left(\frac{\phi_1}{2}\right)}^{2}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right) \]

   3. unpow2 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   4. 1-sub-sin 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   5. pow2 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\cos \left(\frac{\phi_1}{2}\right)}^{2}} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   6. div-inv 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   7. metadata-eval 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot \color{blue}{0.5}\right)}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   8. associate-*r* 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)\right)}}\right) \]

   9. associate-*l* 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}\right)\right)}}\right) \]

   10. pow2 49.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)\right)\right)}}\right) \]

5. Applied egg-rr 49.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)\right)\right)}}}\right) \]
6. Step-by-step derivation

   1. associate-*r* 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}\right)\right)}}\right) \]

   2. *-commutative 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

7. Simplified 49.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)\right)}}}\right) \]

8. Taylor expanded in phi2 around 0 50.4%

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

   1. *-commutative 50.4%

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

10. Simplified 50.4%

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

    1. *-commutative 50.4%

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

    2. metadata-eval 50.4%

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

    3. div-inv 50.4%

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

    4. pow2 50.4%

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

    5. sin-mult 50.4%

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

    6. div-inv 50.4%

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

    7. metadata-eval 50.4%

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

    8. div-inv 50.4%

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

    9. metadata-eval 50.4%

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

    10. cos-sum 50.4%

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

    11. cos-2 50.4%

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

    12. div-inv 50.4%

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

    13. metadata-eval 50.4%

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

12. Applied egg-rr 50.4%

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

    1. div-sub 50.4%

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

    2. +-inverses 50.4%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}{2}\right)}}\right) \]

    3. cos-0 50.4%

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

    4. metadata-eval 50.4%

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

    5. *-commutative 50.4%

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

14. Simplified 50.4%

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

15. Final simplification 50.4%

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

Alternative 16: 46.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = cos(phi1) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)
    code = r * (2.0d0 * atan2(sqrt((t_0 + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt(((cos((phi1 * 0.5d0)) ** 2.0d0) - t_0))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi1) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	return R * (2.0 * math.atan2(math.sqrt((t_0 + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((math.pow(math.cos((phi1 * 0.5)), 2.0) - t_0))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi1) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0);
	tmp = R * (2.0 * atan2(sqrt((t_0 + (sin((phi1 * 0.5)) ^ 2.0))), sqrt(((cos((phi1 * 0.5)) ^ 2.0) - t_0))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

   1. expm1-log1p-u 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(1 - \left({\sin \left(\frac{\phi_1}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)\right)}}}\right) \]

   2. associate--r+ 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\left(1 - {\sin \left(\frac{\phi_1}{2}\right)}^{2}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}\right)\right)}}\right) \]

   3. unpow2 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(1 - \color{blue}{\sin \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_1}{2}\right)}\right) - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   4. 1-sub-sin 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_1}{2}\right)} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   5. pow2 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{{\cos \left(\frac{\phi_1}{2}\right)}^{2}} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   6. div-inv 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \color{blue}{\left(\phi_1 \cdot \frac{1}{2}\right)}}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   7. metadata-eval 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot \color{blue}{0.5}\right)}^{2} - \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}}\right) \]

   8. associate-*r* 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}\right)\right)}}\right) \]

   9. associate-*l* 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)\right)}\right)\right)}}\right) \]

   10. pow2 49.8%

       \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \color{blue}{{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}^{2}}\right)\right)\right)}}\right) \]

5. Applied egg-rr 49.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)\right)\right)}}}\right) \]
6. Step-by-step derivation

   1. associate-*r* 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \color{blue}{\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}\right)\right)}}\right) \]

   2. *-commutative 49.8%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \color{blue}{\left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}^{2}\right)\right)}}\right) \]

7. Simplified 49.8%

   \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\right)\right)}}}\right) \]

8. Taylor expanded in phi2 around 0 50.4%

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

   1. *-commutative 50.4%

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

10. Simplified 50.4%

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

11. Taylor expanded in phi2 around 0 48.6%

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

12. Final simplification 48.6%

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

Alternative 17: 34.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    code = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (cos(phi1) * t_0))), sqrt((1.0d0 - (cos(phi2) * t_0)))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	return R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (math.cos(phi1) * t_0))), math.sqrt((1.0 - (math.cos(phi2) * t_0)))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (cos(phi1) * t_0))), sqrt((1.0 - (cos(phi2) * t_0)))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

4. Taylor expanded in phi1 around 0 35.6%

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

5. Taylor expanded in phi2 around 0 36.2%

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

6. Final simplification 36.2%

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

Alternative 18: 32.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)
    code = r * (2.0d0 * atan2(sqrt((t_0 + (sin((phi2 * (-0.5d0))) ** 2.0d0))), sqrt((1.0d0 - t_0))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	return R * (2.0 * math.atan2(math.sqrt((t_0 + math.pow(math.sin((phi2 * -0.5)), 2.0))), math.sqrt((1.0 - t_0))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0);
	tmp = R * (2.0 * atan2(sqrt((t_0 + (sin((phi2 * -0.5)) ^ 2.0))), sqrt((1.0 - t_0))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

4. Taylor expanded in phi1 around 0 35.6%

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

5. Taylor expanded in phi1 around 0 34.4%

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

6. Final simplification 34.4%

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

Alternative 19: 32.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = (lambda1 - lambda2) * 0.5d0
    code = r * (2.0d0 * atan2(sqrt(((cos(phi1) * (sin(t_0) ** 2.0d0)) + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt((1.0d0 + (cos(phi2) * ((cos((2.0d0 * t_0)) / 2.0d0) - 0.5d0))))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (lambda1 - lambda2) * 0.5
	return R * (2.0 * math.atan2(math.sqrt(((math.cos(phi1) * math.pow(math.sin(t_0), 2.0)) + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((1.0 + (math.cos(phi2) * ((math.cos((2.0 * t_0)) / 2.0) - 0.5))))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (lambda1 - lambda2) * 0.5;
	tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * (sin(t_0) ^ 2.0)) + (sin((phi1 * 0.5)) ^ 2.0))), sqrt((1.0 + (cos(phi2) * ((cos((2.0 * t_0)) / 2.0) - 0.5))))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

4. Taylor expanded in phi1 around 0 35.6%

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

5. Taylor expanded in phi2 around 0 34.1%

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

   1. *-commutative 50.4%

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

   2. metadata-eval 50.4%

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

   3. div-inv 50.4%

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

   4. pow2 50.4%

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

   5. sin-mult 50.4%

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

   6. div-inv 50.4%

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

   7. metadata-eval 50.4%

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

   8. div-inv 50.4%

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

   9. metadata-eval 50.4%

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

   10. cos-sum 50.4%

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

   11. cos-2 50.4%

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

   12. div-inv 50.4%

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

   13. metadata-eval 50.4%

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

7. Applied egg-rr 34.2%

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

   1. div-sub 50.4%

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

   2. +-inverses 50.4%

      \[\leadsto R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot \left(\frac{\cos \color{blue}{0}}{2} - \frac{\cos \left(2 \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\right)}{2}\right)}}\right) \]

   3. cos-0 50.4%

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

   4. metadata-eval 50.4%

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

   5. *-commutative 50.4%

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

9. Simplified 34.2%

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

10. Final simplification 34.2%

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

Alternative 20: 32.6% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    code = r * (2.0d0 * atan2(sqrt(((cos(phi1) * t_0) + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt((1.0d0 - t_0))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	return R * (2.0 * math.atan2(math.sqrt(((math.cos(phi1) * t_0) + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((1.0 - t_0))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * t_0) + (sin((phi1 * 0.5)) ^ 2.0))), sqrt((1.0 - t_0))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

4. Taylor expanded in phi1 around 0 35.6%

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

5. Taylor expanded in phi2 around 0 34.1%

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

6. Taylor expanded in phi2 around 0 34.1%

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

7. Final simplification 34.1%

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

Alternative 21: 16.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    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
    t_0 = sin(((lambda1 - lambda2) * 0.5d0))
    code = r * (2.0d0 * atan2(t_0, sqrt((1.0d0 - (cos(phi2) * (t_0 ** 2.0d0))))))
end function

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) * 0.5))
	return R * (2.0 * math.atan2(t_0, math.sqrt((1.0 - (math.cos(phi2) * math.pow(t_0, 2.0))))))

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) * 0.5));
	tmp = R * (2.0 * atan2(t_0, sqrt((1.0 - (cos(phi2) * (t_0 ^ 2.0))))));
end

Wolfram

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

Derivation

1. Initial program 62.6%

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

3. Taylor expanded in phi1 around inf 49.8%

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

4. Taylor expanded in phi1 around 0 35.6%

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

5. Taylor expanded in phi2 around 0 34.1%

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

6. Taylor expanded in phi1 around 0 18.3%

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

7. Final simplification 18.3%

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

Reproduce

herbie shell --seed 2024157 
(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))))))))))