Distance on a great circle

Percentage Accurate: 61.9% → 79.0%

Time: 55.8s

Alternatives: 19

Speedup: 1.1×

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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: 61.9% 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: 79.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

4. Applied egg-rr 65.6%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

6. Applied egg-rr 81.9%

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

   1. *-commutative 81.9%

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

   2. *-commutative 81.9%

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

   3. fmm-def 81.9%

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

   4. *-commutative 81.9%

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

   5. *-commutative 81.9%

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

   6. *-commutative 81.9%

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

   7. distribute-lft-neg-in 81.9%

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

   8. sin-neg 81.9%

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

   9. distribute-rgt-neg-in 81.9%

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

   10. metadata-eval 81.9%

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

   11. *-commutative 81.9%

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

   12. *-commutative 81.9%

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

8. Simplified 81.9%

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

9. Final simplification 81.9%

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

Alternative 2: 79.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = (t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + pow(((cos((0.5 * phi2)) * sin((phi1 * 0.5))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0);
	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 = (t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (((cos((0.5d0 * phi2)) * sin((phi1 * 0.5d0))) - (cos((phi1 * 0.5d0)) * sin((0.5d0 * phi2)))) ** 2.0d0)
    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 = (t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0)) + Math.pow(((Math.cos((0.5 * phi2)) * Math.sin((phi1 * 0.5))) - (Math.cos((phi1 * 0.5)) * Math.sin((0.5 * phi2)))), 2.0);
	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 = (t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)) + math.pow(((math.cos((0.5 * phi2)) * math.sin((phi1 * 0.5))) - (math.cos((phi1 * 0.5)) * math.sin((0.5 * phi2)))), 2.0)
	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(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(phi1 * 0.5))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0))
	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 = (t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (((cos((0.5 * phi2)) * sin((phi1 * 0.5))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))) ^ 2.0);
	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[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $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]]]

Derivation

1. Initial program 64.5%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

4. Applied egg-rr 65.6%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

6. Applied egg-rr 81.9%

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

7. Final simplification 81.9%

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

Alternative 3: 79.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi1 * 0.5));
	double t_1 = cos((0.5 * phi2)) * sin((phi1 * 0.5));
	double t_2 = cos(phi1) * (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0));
	return 2.0 * (R * atan2(sqrt((t_2 + pow((t_1 - (t_0 * sin((0.5 * phi2)))), 2.0))), sqrt((1.0 - (t_2 + pow((t_1 + (t_0 * sin((phi2 * -0.5)))), 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
    real(8) :: t_1
    real(8) :: t_2
    t_0 = cos((phi1 * 0.5d0))
    t_1 = cos((0.5d0 * phi2)) * sin((phi1 * 0.5d0))
    t_2 = cos(phi1) * (cos(phi2) * (sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0))
    code = 2.0d0 * (r * atan2(sqrt((t_2 + ((t_1 - (t_0 * sin((0.5d0 * phi2)))) ** 2.0d0))), sqrt((1.0d0 - (t_2 + ((t_1 + (t_0 * sin((phi2 * (-0.5d0))))) ** 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 * 0.5));
	double t_1 = Math.cos((0.5 * phi2)) * Math.sin((phi1 * 0.5));
	double t_2 = Math.cos(phi1) * (Math.cos(phi2) * Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0));
	return 2.0 * (R * Math.atan2(Math.sqrt((t_2 + Math.pow((t_1 - (t_0 * Math.sin((0.5 * phi2)))), 2.0))), Math.sqrt((1.0 - (t_2 + Math.pow((t_1 + (t_0 * Math.sin((phi2 * -0.5)))), 2.0))))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

4. Applied egg-rr 65.6%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

6. Applied egg-rr 81.9%

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

   1. *-commutative 81.9%

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

   2. *-commutative 81.9%

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

   3. fmm-def 81.9%

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

   4. *-commutative 81.9%

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

   5. *-commutative 81.9%

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

   6. *-commutative 81.9%

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

   7. distribute-lft-neg-in 81.9%

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

   8. sin-neg 81.9%

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

   9. distribute-rgt-neg-in 81.9%

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

   10. metadata-eval 81.9%

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

   11. *-commutative 81.9%

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

   12. *-commutative 81.9%

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

8. Simplified 81.9%

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

   1. expm1-log1p-u 81.9%

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

10. Applied egg-rr 81.9%

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

11. Taylor expanded in R around 0 81.9%

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

12. Final simplification 81.9%

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

Alternative 4: 62.9% accurate, 0.9× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* t_0 (* (* (cos phi1) (cos phi2)) t_0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        t_1
        (pow
         (-
          (* (cos (* 0.5 phi2)) (sin (* phi1 0.5)))
          (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
         2.0)))
      (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 = t_0 * ((cos(phi1) * cos(phi2)) * t_0);
	return R * (2.0 * atan2(sqrt((t_1 + pow(((cos((0.5 * phi2)) * sin((phi1 * 0.5))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))), 2.0))), 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 = t_0 * ((cos(phi1) * cos(phi2)) * t_0)
    code = r * (2.0d0 * atan2(sqrt((t_1 + (((cos((0.5d0 * phi2)) * sin((phi1 * 0.5d0))) - (cos((phi1 * 0.5d0)) * sin((0.5d0 * phi2)))) ** 2.0d0))), 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 = t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0);
	return R * (2.0 * Math.atan2(Math.sqrt((t_1 + Math.pow(((Math.cos((0.5 * phi2)) * Math.sin((phi1 * 0.5))) - (Math.cos((phi1 * 0.5)) * Math.sin((0.5 * phi2)))), 2.0))), 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 = t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)
	return R * (2.0 * math.atan2(math.sqrt((t_1 + math.pow(((math.cos((0.5 * phi2)) * math.sin((phi1 * 0.5))) - (math.cos((phi1 * 0.5)) * math.sin((0.5 * phi2)))), 2.0))), 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(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + (Float64(Float64(cos(Float64(0.5 * phi2)) * sin(Float64(phi1 * 0.5))) - Float64(cos(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2)))) ^ 2.0))), sqrt(Float64(1.0 + Float64(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 = t_0 * ((cos(phi1) * cos(phi2)) * t_0);
	tmp = R * (2.0 * atan2(sqrt((t_1 + (((cos((0.5 * phi2)) * sin((phi1 * 0.5))) - (cos((phi1 * 0.5)) * sin((0.5 * phi2)))) ^ 2.0))), 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[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[Power[N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 + N[(N[(N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision] - 0.5), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 64.5%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

4. Applied egg-rr 65.6%

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

   1. unpow2 65.6%

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

   2. sin-mult 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

6. Applied egg-rr 65.6%

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

   1. div-sub 65.6%

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

   2. +-inverses 65.6%

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

   3. cos-0 65.6%

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

   4. metadata-eval 65.6%

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

   5. distribute-lft-out 65.6%

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

   6. metadata-eval 65.6%

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

   7. *-rgt-identity 65.6%

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

8. Simplified 65.6%

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

9. Final simplification 65.6%

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

Alternative 5: 62.9% accurate, 0.9× speedup

Math

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

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
       (+
        (* t_1 (* t_0 t_1))
        (pow
         (-
          (* (cos (* 0.5 phi2)) (sin (* phi1 0.5)))
          (* (cos (* phi1 0.5)) (sin (* 0.5 phi2))))
         2.0)))
      (sqrt
       (+
        (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (* t_0 (- (* 0.5 (cos (- lambda1 lambda2))) 0.5)))))))))

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.5%

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

   1. div-sub 64.5%

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

   2. sin-diff 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

4. Applied egg-rr 65.6%

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

   1. associate--r+ 65.6%

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

   2. associate-*l* 65.6%

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

   3. cancel-sign-sub-inv 65.6%

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

   4. div-inv 65.6%

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

   5. metadata-eval 65.6%

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

   6. sqr-sin-a 65.6%

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

   7. cos-2 65.6%

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

   8. cos-sum 65.6%

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

6. Applied egg-rr 65.6%

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

7. Final simplification 65.6%

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

Alternative 6: 62.4% accurate, 0.9× speedup

Math

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

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 (+ (* t_1 (* t_0 t_1)) (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
      (sqrt
       (fabs
        (-
         1.0
         (fma
          t_0
          (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))
          (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(((t_1 * (t_0 * t_1)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt(fabs((1.0 - fma(t_0, (0.5 + (-0.5 * cos((lambda1 - lambda2)))), 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(Float64(t_1 * Float64(t_0 * t_1)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(abs(Float64(1.0 - fma(t_0, Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), (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[(t$95$1 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Abs[N[(1.0 - N[(t$95$0 * N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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 64.5%

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

4. Applied egg-rr 64.8%

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

   1. unpow1/2 64.8%

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

   2. unpow2 64.8%

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

   3. rem-sqrt-square 64.8%

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

   4. cancel-sign-sub-inv 64.8%

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

   5. metadata-eval 64.8%

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

   6. *-commutative 64.8%

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

6. Simplified 64.8%

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

7. Final simplification 64.8%

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

Alternative 7: 62.0% 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 64.5%

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

   1. associate-*l* 64.5%

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

3. Simplified 64.5%

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

   1. unpow2 65.6%

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

   2. sin-mult 65.6%

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

   3. div-inv 65.6%

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

   4. metadata-eval 65.6%

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

   5. div-inv 65.6%

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

   6. metadata-eval 65.6%

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

   7. div-inv 65.6%

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

   8. metadata-eval 65.6%

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

   9. div-inv 65.6%

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

   10. metadata-eval 65.6%

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

6. Applied egg-rr 64.5%

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

   1. div-sub 65.6%

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

   2. +-inverses 65.6%

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

   3. cos-0 65.6%

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

   4. metadata-eval 65.6%

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

   5. distribute-lft-out 65.6%

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

   6. metadata-eval 65.6%

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

   7. *-rgt-identity 65.6%

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

8. Simplified 64.5%

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

9. Final simplification 64.5%

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

Alternative 8: 62.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -3.8 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 9\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot t\_0 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - t\_1 \cdot \left(t\_2 \cdot t\_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 \cdot \left(t\_1 \cdot t\_2\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t\_0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= phi2 -3.8e-5) (not (<= phi2 9.0)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi2) t_0) (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- (pow (cos (* 0.5 phi2)) 2.0) (* t_1 (* t_2 t_2)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* t_2 (* t_1 t_2)) (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- (pow (cos (* phi1 0.5)) 2.0) (* (cos phi1) t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -3.8e-5) || !(phi2 <= 9.0)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * t_0) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((0.5 * phi2)), 2.0) - (t_1 * (t_2 * t_2))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(((t_2 * (t_1 * t_2)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((pow(cos((phi1 * 0.5)), 2.0) - (cos(phi1) * t_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((0.5d0 * (lambda1 - lambda2))) ** 2.0d0
    t_1 = cos(phi1) * cos(phi2)
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    if ((phi2 <= (-3.8d-5)) .or. (.not. (phi2 <= 9.0d0))) then
        tmp = r * (2.0d0 * atan2(sqrt(((cos(phi2) * t_0) + (sin((phi2 * (-0.5d0))) ** 2.0d0))), sqrt(((cos((0.5d0 * phi2)) ** 2.0d0) - (t_1 * (t_2 * t_2))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((t_2 * (t_1 * t_2)) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt(((cos((phi1 * 0.5d0)) ** 2.0d0) - (cos(phi1) * t_0)))))
    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.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -3.8e-5) || !(phi2 <= 9.0)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((Math.cos(phi2) * t_0) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))), Math.sqrt((Math.pow(Math.cos((0.5 * phi2)), 2.0) - (t_1 * (t_2 * t_2))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_2 * (t_1 * t_2)) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((Math.pow(Math.cos((phi1 * 0.5)), 2.0) - (Math.cos(phi1) * t_0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)
	t_1 = math.cos(phi1) * math.cos(phi2)
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if (phi2 <= -3.8e-5) or not (phi2 <= 9.0):
		tmp = R * (2.0 * math.atan2(math.sqrt(((math.cos(phi2) * t_0) + math.pow(math.sin((phi2 * -0.5)), 2.0))), math.sqrt((math.pow(math.cos((0.5 * phi2)), 2.0) - (t_1 * (t_2 * t_2))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_2 * (t_1 * t_2)) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((math.pow(math.cos((phi1 * 0.5)), 2.0) - (math.cos(phi1) * t_0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((phi2 <= -3.8e-5) || !(phi2 <= 9.0))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * t_0) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(0.5 * phi2)) ^ 2.0) - Float64(t_1 * Float64(t_2 * t_2)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_2 * Float64(t_1 * t_2)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64((cos(Float64(phi1 * 0.5)) ^ 2.0) - Float64(cos(phi1) * t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * (lambda1 - lambda2))) ^ 2.0;
	t_1 = cos(phi1) * cos(phi2);
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if ((phi2 <= -3.8e-5) || ~((phi2 <= 9.0)))
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * t_0) + (sin((phi2 * -0.5)) ^ 2.0))), sqrt(((cos((0.5 * phi2)) ^ 2.0) - (t_1 * (t_2 * t_2))))));
	else
		tmp = R * (2.0 * atan2(sqrt(((t_2 * (t_1 * t_2)) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(((cos((phi1 * 0.5)) ^ 2.0) - (cos(phi1) * t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(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, -3.8e-5], N[Not[LessEqual[phi2, 9.0]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$1 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$2 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.8000000000000002e-5 or 9 < phi2

   1. Initial program 43.9%

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

      1. associate-*l* 43.9%

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

   3. Simplified 43.9%

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

   5. Taylor expanded in phi1 around 0 43.8%

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

      1. unpow2 43.8%

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

      2. 1-sub-sin 43.9%

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

      3. unpow2 43.9%

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

      4. *-commutative 43.9%

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

      5. metadata-eval 43.9%

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

      6. distribute-rgt-neg-in 43.9%

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

      7. cos-neg 43.9%

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

   7. Simplified 43.9%

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

   8. Taylor expanded in phi1 around 0 45.2%

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

   if -3.8000000000000002e-5 < phi2 < 9

   1. Initial program 84.7%

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

   3. Taylor expanded in phi2 around 0 84.7%

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

      1. +-commutative 84.7%

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

      2. associate--r+ 84.7%

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

      3. unpow2 84.7%

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

      4. 1-sub-sin 84.8%

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

      5. unpow2 84.8%

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

      6. *-commutative 84.8%

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

   5. Simplified 84.8%

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

4. Final simplification 65.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.8 \cdot 10^{-5} \lor \neg \left(\phi_2 \leq 9\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.9% accurate, 1.1× speedup

Math

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

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 (+ (* t_1 (* t_0 t_1)) (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
      (sqrt
       (+
        (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))
        (* t_0 (- (* 0.5 (cos (- lambda1 lambda2))) 0.5)))))))))

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 64.5%

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

   1. associate--r+ 65.6%

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

   2. associate-*l* 65.6%

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

   3. cancel-sign-sub-inv 65.6%

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

   4. div-inv 65.6%

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

   5. metadata-eval 65.6%

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

   6. sqr-sin-a 65.6%

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

   7. cos-2 65.6%

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

   8. cos-sum 65.6%

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

4. Applied egg-rr 64.5%

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

5. Final simplification 64.5%

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

Alternative 10: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-28} \lor \neg \left(\phi_2 \leq 6.2 \cdot 10^{-45}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot t\_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot t\_1 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{t\_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (sqrt
          (-
           1.0
           (fma
            (* (cos phi1) (cos phi2))
            (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))
            (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)))
   (if (or (<= phi2 -1.95e-28) (not (<= phi2 6.2e-45)))
     (*
      (* R 2.0)
      (atan2 (sqrt (+ (* (cos phi2) t_1) (pow (sin (* phi2 -0.5)) 2.0))) t_0))
     (*
      (* R 2.0)
      (atan2
       (sqrt (+ (* (cos phi1) t_1) (pow (sin (* phi1 0.5)) 2.0)))
       t_0)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sqrt((1.0 - fma((cos(phi1) * cos(phi2)), (0.5 + (-0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * (phi1 - phi2))), 2.0))));
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double tmp;
	if ((phi2 <= -1.95e-28) || !(phi2 <= 6.2e-45)) {
		tmp = (R * 2.0) * atan2(sqrt(((cos(phi2) * t_1) + pow(sin((phi2 * -0.5)), 2.0))), t_0);
	} else {
		tmp = (R * 2.0) * atan2(sqrt(((cos(phi1) * t_1) + pow(sin((phi1 * 0.5)), 2.0))), t_0);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt(Float64(1.0 - fma(Float64(cos(phi1) * cos(phi2)), Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -1.95e-28) || !(phi2 <= 6.2e-45))
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(Float64(Float64(cos(phi2) * t_1) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), t_0));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(Float64(Float64(cos(phi1) * t_1) + (sin(Float64(phi1 * 0.5)) ^ 2.0))), t_0));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = 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[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -1.95e-28], N[Not[LessEqual[phi2, 6.2e-45]], $MachinePrecision]], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[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] / t$95$0], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.94999999999999999e-28 or 6.2000000000000002e-45 < phi2

   1. Initial program 44.7%

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

      1. associate-*r* 44.7%

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

      2. *-commutative 44.7%

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

   3. Simplified 44.7%

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

   6. Applied egg-rr 22.0%

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

      1. *-lft-identity 22.0%

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

      2. *-commutative 22.0%

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

      3. *-commutative 22.0%

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

      4. cancel-sign-sub-inv 22.0%

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

      5. metadata-eval 22.0%

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

   8. Simplified 22.0%

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

   9. Taylor expanded in phi1 around 0 44.9%

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

   if -1.94999999999999999e-28 < phi2 < 6.2000000000000002e-45

   1. Initial program 87.6%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   6. Applied egg-rr 59.6%

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

      1. *-lft-identity 59.6%

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

      2. *-commutative 59.6%

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

      3. *-commutative 59.6%

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

      4. cancel-sign-sub-inv 59.6%

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

      5. metadata-eval 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in phi2 around 0 58.8%

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

   10. Taylor expanded in phi2 around 0 86.9%

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

4. Final simplification 64.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-28} \lor \neg \left(\phi_2 \leq 6.2 \cdot 10^{-45}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-28} \lor \neg \left(\phi_2 \leq 6.2 \cdot 10^{-45}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot t\_0 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - t\_1 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot t\_0 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(t\_1, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (* (cos phi1) (cos phi2))))
   (if (or (<= phi2 -1.95e-28) (not (<= phi2 6.2e-45)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi2) t_0) (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt
         (-
          (pow (cos (* 0.5 phi2)) 2.0)
          (*
           t_1
           (* (sin (/ (- lambda1 lambda2) 2.0)) (sin (* 0.5 lambda1)))))))))
     (*
      (* R 2.0)
      (atan2
       (sqrt (+ (* (cos phi1) t_0) (pow (sin (* phi1 0.5)) 2.0)))
       (sqrt
        (-
         1.0
         (fma
          t_1
          (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if ((phi2 <= -1.95e-28) || !(phi2 <= 6.2e-45)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * t_0) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((0.5 * phi2)), 2.0) - (t_1 * (sin(((lambda1 - lambda2) / 2.0)) * sin((0.5 * lambda1))))))));
	} else {
		tmp = (R * 2.0) * atan2(sqrt(((cos(phi1) * t_0) + pow(sin((phi1 * 0.5)), 2.0))), sqrt((1.0 - fma(t_1, (0.5 + (-0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if ((phi2 <= -1.95e-28) || !(phi2 <= 6.2e-45))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * t_0) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(0.5 * phi2)) ^ 2.0) - Float64(t_1 * Float64(sin(Float64(Float64(lambda1 - lambda2) / 2.0)) * sin(Float64(0.5 * lambda1)))))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(Float64(Float64(cos(phi1) * t_0) + (sin(Float64(phi1 * 0.5)) ^ 2.0))), sqrt(Float64(1.0 - fma(t_1, Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -1.95e-28], N[Not[LessEqual[phi2, 6.2e-45]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$1 * N[(N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * 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 - N[(t$95$1 * N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.94999999999999999e-28 or 6.2000000000000002e-45 < phi2

   1. Initial program 44.7%

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

      1. associate-*l* 44.7%

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

   3. Simplified 44.7%

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

   5. Taylor expanded in phi1 around 0 44.2%

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

      1. unpow2 44.2%

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

      2. 1-sub-sin 44.3%

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

      3. unpow2 44.3%

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

      4. *-commutative 44.3%

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

      5. metadata-eval 44.3%

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

      6. distribute-rgt-neg-in 44.3%

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

      7. cos-neg 44.3%

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

   7. Simplified 44.3%

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

   8. Taylor expanded in lambda2 around 0 36.6%

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

   9. Taylor expanded in phi1 around 0 37.6%

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

   if -1.94999999999999999e-28 < phi2 < 6.2000000000000002e-45

   1. Initial program 87.6%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   6. Applied egg-rr 59.6%

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

      1. *-lft-identity 59.6%

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

      2. *-commutative 59.6%

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

      3. *-commutative 59.6%

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

      4. cancel-sign-sub-inv 59.6%

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

      5. metadata-eval 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in phi2 around 0 58.8%

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

   10. Taylor expanded in phi2 around 0 86.9%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-28} \lor \neg \left(\phi_2 \leq 6.2 \cdot 10^{-45}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 48.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\phi_2 \leq -0.00215 \lor \neg \left(\phi_2 \leq 0.00012\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {t\_0}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - t\_1 \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), t\_0 \cdot \sqrt{t\_1}\right)}{\sqrt{1 - \mathsf{fma}\left(t\_1, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* 0.5 (- lambda1 lambda2))))
        (t_1 (* (cos phi1) (cos phi2))))
   (if (or (<= phi2 -0.00215) (not (<= phi2 0.00012)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi2) (pow t_0 2.0)) (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt
         (-
          (pow (cos (* 0.5 phi2)) 2.0)
          (*
           t_1
           (* (sin (/ (- lambda1 lambda2) 2.0)) (sin (* 0.5 lambda1)))))))))
     (*
      (* R 2.0)
      (atan2
       (hypot (sin (* 0.5 (- phi1 phi2))) (* t_0 (sqrt t_1)))
       (sqrt
        (-
         1.0
         (fma
          t_1
          (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))
          (pow (sin (* phi1 0.5)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * (lambda1 - lambda2)));
	double t_1 = cos(phi1) * cos(phi2);
	double tmp;
	if ((phi2 <= -0.00215) || !(phi2 <= 0.00012)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * pow(t_0, 2.0)) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((0.5 * phi2)), 2.0) - (t_1 * (sin(((lambda1 - lambda2) / 2.0)) * sin((0.5 * lambda1))))))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(sin((0.5 * (phi1 - phi2))), (t_0 * sqrt(t_1))), sqrt((1.0 - fma(t_1, (0.5 + (-0.5 * cos((lambda1 - lambda2)))), pow(sin((phi1 * 0.5)), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_1 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if ((phi2 <= -0.00215) || !(phi2 <= 0.00012))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * (t_0 ^ 2.0)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(0.5 * phi2)) ^ 2.0) - Float64(t_1 * Float64(sin(Float64(Float64(lambda1 - lambda2) / 2.0)) * sin(Float64(0.5 * lambda1)))))))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(0.5 * Float64(phi1 - phi2))), Float64(t_0 * sqrt(t_1))), sqrt(Float64(1.0 - fma(t_1, Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(phi1 * 0.5)) ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi2, -0.00215], N[Not[LessEqual[phi2, 0.00012]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$1 * N[(N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * lambda1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $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$1], $MachinePrecision]), $MachinePrecision] ^ 2], $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[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -0.00215 or 1.20000000000000003e-4 < phi2

   1. Initial program 43.6%

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

      1. associate-*l* 43.6%

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

   3. Simplified 43.6%

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

   5. Taylor expanded in phi1 around 0 43.4%

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

      1. unpow2 43.4%

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

      2. 1-sub-sin 43.6%

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

      3. unpow2 43.6%

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

      4. *-commutative 43.6%

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

      5. metadata-eval 43.6%

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

      6. distribute-rgt-neg-in 43.6%

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

      7. cos-neg 43.6%

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

   7. Simplified 43.6%

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

   8. Taylor expanded in lambda2 around 0 36.0%

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

   9. Taylor expanded in phi1 around 0 37.0%

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

   if -0.00215 < phi2 < 1.20000000000000003e-4

   1. Initial program 85.7%

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

      1. associate-*r* 85.7%

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

      2. *-commutative 85.7%

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

   3. Simplified 85.7%

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

   6. Applied egg-rr 59.5%

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

      1. *-lft-identity 59.5%

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

      2. *-commutative 59.5%

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

      3. *-commutative 59.5%

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

      4. cancel-sign-sub-inv 59.5%

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

      5. metadata-eval 59.5%

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

   8. Simplified 59.5%

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

   9. Taylor expanded in phi2 around 0 59.5%

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

4. Final simplification 48.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -0.00215 \lor \neg \left(\phi_2 \leq 0.00012\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(0.5 \cdot \lambda_1\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sqrt{1 - \mathsf{fma}\left(t\_1, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_4 := \sqrt{\cos \phi_2 \cdot t\_0 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_4}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - t\_1 \cdot \left(t\_3 \cdot t\_3\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot t\_0 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{t\_4}{t\_2}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2
         (sqrt
          (-
           1.0
           (fma
            t_1
            (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))
            (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_4 (sqrt (+ (* (cos phi2) t_0) (pow (sin (* phi2 -0.5)) 2.0)))))
   (if (<= phi2 -1.95e-28)
     (*
      R
      (*
       2.0
       (atan2
        t_4
        (sqrt (- (pow (cos (* 0.5 phi2)) 2.0) (* t_1 (* t_3 t_3)))))))
     (if (<= phi2 6.2e-45)
       (*
        (* R 2.0)
        (atan2 (sqrt (+ (* (cos phi1) t_0) (pow (sin (* phi1 0.5)) 2.0))) t_2))
       (* (* R 2.0) (atan2 t_4 t_2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sqrt((1.0 - fma(t_1, (0.5 + (-0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * (phi1 - phi2))), 2.0))));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double t_4 = sqrt(((cos(phi2) * t_0) + pow(sin((phi2 * -0.5)), 2.0)));
	double tmp;
	if (phi2 <= -1.95e-28) {
		tmp = R * (2.0 * atan2(t_4, sqrt((pow(cos((0.5 * phi2)), 2.0) - (t_1 * (t_3 * t_3))))));
	} else if (phi2 <= 6.2e-45) {
		tmp = (R * 2.0) * atan2(sqrt(((cos(phi1) * t_0) + pow(sin((phi1 * 0.5)), 2.0))), t_2);
	} else {
		tmp = (R * 2.0) * atan2(t_4, t_2);
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = sqrt(Float64(1.0 - fma(t_1, Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_4 = sqrt(Float64(Float64(cos(phi2) * t_0) + (sin(Float64(phi2 * -0.5)) ^ 2.0)))
	tmp = 0.0
	if (phi2 <= -1.95e-28)
		tmp = Float64(R * Float64(2.0 * atan(t_4, sqrt(Float64((cos(Float64(0.5 * phi2)) ^ 2.0) - Float64(t_1 * Float64(t_3 * t_3)))))));
	elseif (phi2 <= 6.2e-45)
		tmp = Float64(Float64(R * 2.0) * atan(sqrt(Float64(Float64(cos(phi1) * t_0) + (sin(Float64(phi1 * 0.5)) ^ 2.0))), t_2));
	else
		tmp = Float64(Float64(R * 2.0) * atan(t_4, t_2));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = 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[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -1.95e-28], N[(R * N[(2.0 * N[ArcTan[t$95$4 / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(t$95$1 * N[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 6.2e-45], N[(N[(R * 2.0), $MachinePrecision] * 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] / t$95$2], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[t$95$4 / t$95$2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -1.94999999999999999e-28

   1. Initial program 45.0%

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

      1. associate-*l* 45.0%

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

   3. Simplified 45.0%

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

   5. Taylor expanded in phi1 around 0 44.8%

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

      1. unpow2 44.8%

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

      2. 1-sub-sin 44.9%

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

      3. unpow2 44.9%

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

      4. *-commutative 44.9%

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

      5. metadata-eval 44.9%

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

      6. distribute-rgt-neg-in 44.9%

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

      7. cos-neg 44.9%

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

   7. Simplified 44.9%

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

   8. Taylor expanded in phi1 around 0 46.3%

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

   if -1.94999999999999999e-28 < phi2 < 6.2000000000000002e-45

   1. Initial program 87.6%

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

      1. associate-*r* 87.6%

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

      2. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   6. Applied egg-rr 59.6%

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

      1. *-lft-identity 59.6%

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

      2. *-commutative 59.6%

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

      3. *-commutative 59.6%

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

      4. cancel-sign-sub-inv 59.6%

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

      5. metadata-eval 59.6%

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

   8. Simplified 59.6%

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

   9. Taylor expanded in phi2 around 0 58.8%

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

   10. Taylor expanded in phi2 around 0 86.9%

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

   if 6.2000000000000002e-45 < phi2

   1. Initial program 44.5%

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

      1. associate-*r* 44.5%

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

      2. *-commutative 44.5%

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

   3. Simplified 44.5%

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

   6. Applied egg-rr 20.4%

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

      1. *-lft-identity 20.4%

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

      2. *-commutative 20.4%

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

      3. *-commutative 20.4%

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

      4. cancel-sign-sub-inv 20.4%

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

      5. metadata-eval 20.4%

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

   8. Simplified 20.4%

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

   9. Taylor expanded in phi1 around 0 44.7%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.95 \cdot 10^{-28}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_2\right)}^{2} - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \mathbf{elif}\;\phi_2 \leq 6.2 \cdot 10^{-45}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 39.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_3 := {t\_2}^{2}\\ \mathbf{if}\;\lambda_2 \leq -0.00015 \lor \neg \left(\lambda_2 \leq 10^{-68}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\phi_1 \cdot 0.5\right), t\_1 \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(t\_0, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_2, t\_1 \cdot \sqrt{t\_0}\right)}{\sqrt{1 - \mathsf{fma}\left(t\_0, 0.5 + -0.5 \cdot \cos \lambda_1, t\_3\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (sin (* 0.5 (- lambda1 lambda2))))
        (t_2 (sin (* 0.5 (- phi1 phi2))))
        (t_3 (pow t_2 2.0)))
   (if (or (<= lambda2 -0.00015) (not (<= lambda2 1e-68)))
     (*
      (* R 2.0)
      (atan2
       (hypot (sin (* phi1 0.5)) (* t_1 (sqrt (cos phi1))))
       (sqrt
        (- 1.0 (fma t_0 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2)))) t_3)))))
     (*
      (* R 2.0)
      (atan2
       (hypot t_2 (* t_1 (sqrt t_0)))
       (sqrt (- 1.0 (fma t_0 (+ 0.5 (* -0.5 (cos lambda1))) t_3))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin((0.5 * (lambda1 - lambda2)));
	double t_2 = sin((0.5 * (phi1 - phi2)));
	double t_3 = pow(t_2, 2.0);
	double tmp;
	if ((lambda2 <= -0.00015) || !(lambda2 <= 1e-68)) {
		tmp = (R * 2.0) * atan2(hypot(sin((phi1 * 0.5)), (t_1 * sqrt(cos(phi1)))), sqrt((1.0 - fma(t_0, (0.5 + (-0.5 * cos((lambda1 - lambda2)))), t_3))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(t_2, (t_1 * sqrt(t_0))), sqrt((1.0 - fma(t_0, (0.5 + (-0.5 * cos(lambda1))), t_3))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_2 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_3 = t_2 ^ 2.0
	tmp = 0.0
	if ((lambda2 <= -0.00015) || !(lambda2 <= 1e-68))
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(phi1 * 0.5)), Float64(t_1 * sqrt(cos(phi1)))), sqrt(Float64(1.0 - fma(t_0, Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), t_3)))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_2, Float64(t_1 * sqrt(t_0))), sqrt(Float64(1.0 - fma(t_0, Float64(0.5 + Float64(-0.5 * cos(lambda1))), t_3)))));
	end
	return tmp
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[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, If[Or[LessEqual[lambda2, -0.00015], N[Not[LessEqual[lambda2, 1e-68]], $MachinePrecision]], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(t$95$1 * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$2 ^ 2 + N[(t$95$1 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[(0.5 + N[(-0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.49999999999999987e-4 or 1.00000000000000007e-68 < lambda2

   1. Initial program 51.9%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   6. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

      2. *-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. cancel-sign-sub-inv 29.6%

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

      5. metadata-eval 29.6%

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

   8. Simplified 29.6%

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

   9. Taylor expanded in phi2 around 0 27.2%

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

   10. Taylor expanded in phi2 around 0 32.2%

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

   if -1.49999999999999987e-4 < lambda2 < 1.00000000000000007e-68

   1. Initial program 79.6%

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

      1. associate-*r* 79.6%

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

      2. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   6. Applied egg-rr 51.1%

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

      1. *-lft-identity 51.1%

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

      2. *-commutative 51.1%

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

      3. *-commutative 51.1%

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

      4. cancel-sign-sub-inv 51.1%

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

      5. metadata-eval 51.1%

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

   8. Simplified 51.1%

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

   9. Taylor expanded in lambda2 around 0 51.1%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.00015 \lor \neg \left(\lambda_2 \leq 10^{-68}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\phi_1 \cdot 0.5\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \lambda_1, {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 61.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((0.5 * (phi1 - phi2))), 2.0);
	return atan2(sqrt(((cos(phi1) * (cos(phi2) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0))) + t_0)), sqrt((1.0 + ((cos(phi1) * (cos(phi2) * ((0.5 * cos((lambda1 - lambda2))) - 0.5))) - t_0)))) * (R * 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((0.5d0 * (phi1 - phi2))) ** 2.0d0
    code = atan2(sqrt(((cos(phi1) * (cos(phi2) * (sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0))) + t_0)), sqrt((1.0d0 + ((cos(phi1) * (cos(phi2) * ((0.5d0 * cos((lambda1 - lambda2))) - 0.5d0))) - t_0)))) * (r * 2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

3. Simplified 64.4%

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

   1. add-cube-cbrt 64.2%

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

   2. pow3 64.2%

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

6. Applied egg-rr 64.3%

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

7. Taylor expanded in phi1 around 0 64.5%

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

8. Final simplification 64.5%

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

Alternative 16: 39.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\\ t_2 := \sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)\\ t_3 := {t\_2}^{2}\\ \mathbf{if}\;\lambda_2 \leq -0.000105 \lor \neg \left(\lambda_2 \leq 1.2 \cdot 10^{-68}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\phi_1 \cdot 0.5\right), t\_1 \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(t\_0, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), t\_3\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(t\_2, t\_1 \cdot \sqrt{t\_0}\right)}{\sqrt{1 - \left(t\_3 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_1\right)\right)\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (sin (* 0.5 (- lambda1 lambda2))))
        (t_2 (sin (* 0.5 (- phi1 phi2))))
        (t_3 (pow t_2 2.0)))
   (if (or (<= lambda2 -0.000105) (not (<= lambda2 1.2e-68)))
     (*
      (* R 2.0)
      (atan2
       (hypot (sin (* phi1 0.5)) (* t_1 (sqrt (cos phi1))))
       (sqrt
        (- 1.0 (fma t_0 (+ 0.5 (* -0.5 (cos (- lambda1 lambda2)))) t_3)))))
     (*
      (* R 2.0)
      (atan2
       (hypot t_2 (* t_1 (sqrt t_0)))
       (sqrt
        (-
         1.0
         (+
          t_3
          (* (cos phi1) (* (cos phi2) (+ 0.5 (* -0.5 (cos lambda1)))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi1) * cos(phi2);
	double t_1 = sin((0.5 * (lambda1 - lambda2)));
	double t_2 = sin((0.5 * (phi1 - phi2)));
	double t_3 = pow(t_2, 2.0);
	double tmp;
	if ((lambda2 <= -0.000105) || !(lambda2 <= 1.2e-68)) {
		tmp = (R * 2.0) * atan2(hypot(sin((phi1 * 0.5)), (t_1 * sqrt(cos(phi1)))), sqrt((1.0 - fma(t_0, (0.5 + (-0.5 * cos((lambda1 - lambda2)))), t_3))));
	} else {
		tmp = (R * 2.0) * atan2(hypot(t_2, (t_1 * sqrt(t_0))), sqrt((1.0 - (t_3 + (cos(phi1) * (cos(phi2) * (0.5 + (-0.5 * cos(lambda1)))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2)))
	t_2 = sin(Float64(0.5 * Float64(phi1 - phi2)))
	t_3 = t_2 ^ 2.0
	tmp = 0.0
	if ((lambda2 <= -0.000105) || !(lambda2 <= 1.2e-68))
		tmp = Float64(Float64(R * 2.0) * atan(hypot(sin(Float64(phi1 * 0.5)), Float64(t_1 * sqrt(cos(phi1)))), sqrt(Float64(1.0 - fma(t_0, Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), t_3)))));
	else
		tmp = Float64(Float64(R * 2.0) * atan(hypot(t_2, Float64(t_1 * sqrt(t_0))), sqrt(Float64(1.0 - Float64(t_3 + Float64(cos(phi1) * Float64(cos(phi2) * Float64(0.5 + Float64(-0.5 * cos(lambda1))))))))));
	end
	return tmp
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[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Power[t$95$2, 2.0], $MachinePrecision]}, If[Or[LessEqual[lambda2, -0.000105], N[Not[LessEqual[lambda2, 1.2e-68]], $MachinePrecision]], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(t$95$1 * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 * N[(0.5 + N[(-0.5 * N[Cos[N[(lambda1 - lambda2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$3), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[t$95$2 ^ 2 + N[(t$95$1 * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$3 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(0.5 + N[(-0.5 * N[Cos[lambda1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < -1.05e-4 or 1.19999999999999996e-68 < lambda2

   1. Initial program 51.9%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

   3. Simplified 51.9%

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

   6. Applied egg-rr 29.6%

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

      1. *-lft-identity 29.6%

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

      2. *-commutative 29.6%

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

      3. *-commutative 29.6%

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

      4. cancel-sign-sub-inv 29.6%

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

      5. metadata-eval 29.6%

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

   8. Simplified 29.6%

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

   9. Taylor expanded in phi2 around 0 27.2%

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

   10. Taylor expanded in phi2 around 0 32.2%

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

   if -1.05e-4 < lambda2 < 1.19999999999999996e-68

   1. Initial program 79.6%

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

      1. associate-*r* 79.6%

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

      2. *-commutative 79.6%

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

   3. Simplified 79.6%

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

   6. Applied egg-rr 51.1%

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

      1. *-lft-identity 51.1%

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

      2. *-commutative 51.1%

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

      3. *-commutative 51.1%

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

      4. cancel-sign-sub-inv 51.1%

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

      5. metadata-eval 51.1%

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

   8. Simplified 51.1%

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

      1. cos-diff 51.1%

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

   10. Applied egg-rr 51.1%

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

   11. Taylor expanded in lambda2 around 0 51.0%

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

4. Final simplification 40.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -0.000105 \lor \neg \left(\lambda_2 \leq 1.2 \cdot 10^{-68}\right):\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(\phi_1 \cdot 0.5\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\mathsf{hypot}\left(\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right), \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \sqrt{\cos \phi_1 \cdot \cos \phi_2}\right)}{\sqrt{1 - \left({\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot \left(0.5 + -0.5 \cdot \cos \lambda_1\right)\right)\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 34.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[Sqrt[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] ^ 2 + N[(N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Cos[phi1], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $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[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.5%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

3. Simplified 64.4%

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

6. Applied egg-rr 39.3%

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

   1. *-lft-identity 39.3%

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

   2. *-commutative 39.3%

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

   3. *-commutative 39.3%

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

   4. cancel-sign-sub-inv 39.3%

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

   5. metadata-eval 39.3%

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

8. Simplified 39.3%

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

9. Taylor expanded in phi2 around 0 32.3%

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

10. Taylor expanded in phi2 around 0 35.6%

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

11. Final simplification 35.6%

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

Alternative 18: 16.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(1 + -0.25 \cdot {\phi_2}^{2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  (* R 2.0)
  (atan2
   (* (sin (* 0.5 (- lambda1 lambda2))) (+ 1.0 (* -0.25 (pow phi2 2.0))))
   (sqrt
    (-
     1.0
     (fma
      (* (cos phi1) (cos phi2))
      (+ 0.5 (* -0.5 (cos (- lambda1 lambda2))))
      (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return (R * 2.0) * atan2((sin((0.5 * (lambda1 - lambda2))) * (1.0 + (-0.25 * pow(phi2, 2.0)))), sqrt((1.0 - fma((cos(phi1) * cos(phi2)), (0.5 + (-0.5 * cos((lambda1 - lambda2)))), pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(Float64(R * 2.0) * atan(Float64(sin(Float64(0.5 * Float64(lambda1 - lambda2))) * Float64(1.0 + Float64(-0.25 * (phi2 ^ 2.0)))), sqrt(Float64(1.0 - fma(Float64(cos(phi1) * cos(phi2)), Float64(0.5 + Float64(-0.5 * cos(Float64(lambda1 - lambda2)))), (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(N[(R * 2.0), $MachinePrecision] * N[ArcTan[N[(N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(-0.25 * N[Power[phi2, 2.0], $MachinePrecision]), $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[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 64.5%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

3. Simplified 64.4%

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

6. Applied egg-rr 39.3%

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

   1. *-lft-identity 39.3%

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

   2. *-commutative 39.3%

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

   3. *-commutative 39.3%

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

   4. cancel-sign-sub-inv 39.3%

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

   5. metadata-eval 39.3%

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

8. Simplified 39.3%

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

9. Taylor expanded in phi2 around 0 32.3%

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

10. Taylor expanded in phi1 around 0 13.2%

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

11. Taylor expanded in phi2 around 0 16.3%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) + -0.25 \cdot \left({\phi_2}^{2} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]
12. Step-by-step derivation

    1. associate-*r* 16.3%

       \[\leadsto \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) + \color{blue}{\left(-0.25 \cdot {\phi_2}^{2}\right) \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    2. distribute-rgt1-in 16.3%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(-0.25 \cdot {\phi_2}^{2} + 1\right) \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

    3. +-commutative 16.3%

       \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.25 \cdot {\phi_2}^{2}\right)} \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

13. Simplified 16.3%

    \[\leadsto \tan^{-1}_* \frac{\color{blue}{\left(1 + -0.25 \cdot {\phi_2}^{2}\right) \cdot \sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \cdot \left(R \cdot 2\right) \]

14. Final simplification 16.3%

    \[\leadsto \left(R \cdot 2\right) \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(1 + -0.25 \cdot {\phi_2}^{2}\right)}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_1 \cdot \cos \phi_2, 0.5 + -0.5 \cdot \cos \left(\lambda_1 - \lambda_2\right), {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}\right)}} \]
15. Add Preprocessing

Alternative 19: 16.5% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

   1. associate-*r* 64.5%

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

   2. *-commutative 64.5%

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

3. Simplified 64.4%

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

6. Applied egg-rr 39.3%

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

   1. *-lft-identity 39.3%

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

   2. *-commutative 39.3%

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

   3. *-commutative 39.3%

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

   4. cancel-sign-sub-inv 39.3%

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

   5. metadata-eval 39.3%

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

8. Simplified 39.3%

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

9. Taylor expanded in phi2 around 0 32.3%

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

10. Taylor expanded in phi1 around 0 13.2%

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

11. Taylor expanded in phi2 around 0 15.4%

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

12. Final simplification 15.4%

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

Reproduce

herbie shell --seed 2024154 
(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))))))))))