Distance on a great circle

Percentage Accurate: 62.1% → 78.3%

Time: 2.4min

Alternatives: 21

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.1% accurate, 1.0× speedup

Math

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

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: 78.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_2\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(0.5 \cdot \phi_1\right)\\ t_3 := \sin \left(0.5 \cdot \phi_2\right)\\ t_4 := \cos \phi_1 \cdot \cos \phi_2\\ t_5 := \sin \left(0.5 \cdot \phi_1\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\left(\mathsf{fma}\left(t\_5, t\_0, t\_2 \cdot \left(-t\_3\right)\right)\right)}^{2} + \left(t\_4 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\right) \cdot t\_1}}{\sqrt{1 - \left(t\_1 \cdot \left(t\_4 \cdot t\_1\right) + {\left(t\_5 \cdot t\_0 - t\_2 \cdot t\_3\right)}^{2}\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi2)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (* 0.5 phi1)))
        (t_3 (sin (* 0.5 phi2)))
        (t_4 (* (cos phi1) (cos phi2)))
        (t_5 (sin (* 0.5 phi1))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (fma t_5 t_0 (* t_2 (- t_3))) 2.0)
        (* (* t_4 (log1p (expm1 (sin (* 0.5 (- lambda1 lambda2)))))) t_1)))
      (sqrt
       (-
        1.0
        (+ (* t_1 (* t_4 t_1)) (pow (- (* t_5 t_0) (* t_2 t_3)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi2));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos((0.5 * phi1));
	double t_3 = sin((0.5 * phi2));
	double t_4 = cos(phi1) * cos(phi2);
	double t_5 = sin((0.5 * phi1));
	return R * (2.0 * atan2(sqrt((pow(fma(t_5, t_0, (t_2 * -t_3)), 2.0) + ((t_4 * log1p(expm1(sin((0.5 * (lambda1 - lambda2)))))) * t_1))), sqrt((1.0 - ((t_1 * (t_4 * t_1)) + pow(((t_5 * t_0) - (t_2 * t_3)), 2.0))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi2))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(0.5 * phi1))
	t_3 = sin(Float64(0.5 * phi2))
	t_4 = Float64(cos(phi1) * cos(phi2))
	t_5 = sin(Float64(0.5 * phi1))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((fma(t_5, t_0, Float64(t_2 * Float64(-t_3))) ^ 2.0) + Float64(Float64(t_4 * log1p(expm1(sin(Float64(0.5 * Float64(lambda1 - lambda2)))))) * t_1))), sqrt(Float64(1.0 - Float64(Float64(t_1 * Float64(t_4 * t_1)) + (Float64(Float64(t_5 * t_0) - Float64(t_2 * t_3)) ^ 2.0)))))))
end

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

4. Applied egg-rr 66.6%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

6. Applied egg-rr 78.0%

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

   1. fma-neg 78.0%

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

   2. *-commutative 78.0%

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

   3. *-commutative 78.0%

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

   4. *-commutative 78.0%

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

   5. *-commutative 78.0%

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

   6. *-commutative 78.0%

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

   7. distribute-rgt-neg-in 78.0%

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

8. Simplified 78.0%

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

   1. add-sqr-sqrt 34.6%

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

   2. sqrt-prod 51.6%

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

   3. log1p-expm1-u 51.6%

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

   4. sqrt-prod 34.6%

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

   5. add-sqr-sqrt 78.0%

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

   6. div-inv 78.0%

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

   7. metadata-eval 78.0%

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

10. Applied egg-rr 78.0%

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

11. Final simplification 78.0%

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

Alternative 2: 78.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

4. Applied egg-rr 66.6%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

6. Applied egg-rr 78.0%

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

   1. fma-neg 78.0%

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

   2. *-commutative 78.0%

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

   3. *-commutative 78.0%

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

   4. *-commutative 78.0%

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

   5. *-commutative 78.0%

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

   6. *-commutative 78.0%

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

   7. distribute-rgt-neg-in 78.0%

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

8. Simplified 78.0%

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

9. Taylor expanded in phi1 around inf 78.0%

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

    1. fma-neg 78.0%

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

    2. *-commutative 78.0%

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

    3. *-commutative 78.0%

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

    4. distribute-lft-neg-in 78.0%

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

    5. sin-neg 78.0%

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

    6. distribute-lft-neg-in 78.0%

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

    7. metadata-eval 78.0%

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

    8. *-commutative 78.0%

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

    9. *-commutative 78.0%

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

11. Simplified 78.0%

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

12. Final simplification 78.0%

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

Alternative 3: 78.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

4. Applied egg-rr 66.6%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

6. Applied egg-rr 78.0%

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

   1. fma-neg 78.0%

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

   2. *-commutative 78.0%

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

   3. *-commutative 78.0%

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

   4. *-commutative 78.0%

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

   5. *-commutative 78.0%

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

   6. *-commutative 78.0%

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

   7. distribute-rgt-neg-in 78.0%

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

8. Simplified 78.0%

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

9. Final simplification 78.0%

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

Alternative 4: 72.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\\ t_2 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\\ t_3 := t\_0 \cdot t\_2\\ t_4 := t\_3 + t\_1\\ \mathbf{if}\;\lambda_1 \leq -1.4 \lor \neg \left(\lambda_1 \leq 3 \cdot 10^{-18}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_3 + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{1 - t\_4}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4}}{\sqrt{1 - \left(t\_1 + t\_2 \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (pow
          (-
           (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0))
        (t_2 (* (* (cos phi1) (cos phi2)) t_0))
        (t_3 (* t_0 t_2))
        (t_4 (+ t_3 t_1)))
   (if (or (<= lambda1 -1.4) (not (<= lambda1 3e-18)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_3 (- 0.5 (/ (cos (- phi1 phi2)) 2.0))))
        (sqrt (- 1.0 t_4)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt t_4)
        (sqrt (- 1.0 (+ t_1 (* t_2 (sin (* lambda2 -0.5))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0);
	double t_2 = (cos(phi1) * cos(phi2)) * t_0;
	double t_3 = t_0 * t_2;
	double t_4 = t_3 + t_1;
	double tmp;
	if ((lambda1 <= -1.4) || !(lambda1 <= 3e-18)) {
		tmp = R * (2.0 * atan2(sqrt((t_3 + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - (t_1 + (t_2 * sin((lambda2 * -0.5))))))));
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = ((sin((0.5d0 * phi1)) * cos((0.5d0 * phi2))) - (cos((0.5d0 * phi1)) * sin((0.5d0 * phi2)))) ** 2.0d0
    t_2 = (cos(phi1) * cos(phi2)) * t_0
    t_3 = t_0 * t_2
    t_4 = t_3 + t_1
    if ((lambda1 <= (-1.4d0)) .or. (.not. (lambda1 <= 3d-18))) then
        tmp = r * (2.0d0 * atan2(sqrt((t_3 + (0.5d0 - (cos((phi1 - phi2)) / 2.0d0)))), sqrt((1.0d0 - t_4))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(t_4), sqrt((1.0d0 - (t_1 + (t_2 * sin((lambda2 * (-0.5d0)))))))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(((Math.sin((0.5 * phi1)) * Math.cos((0.5 * phi2))) - (Math.cos((0.5 * phi1)) * Math.sin((0.5 * phi2)))), 2.0);
	double t_2 = (Math.cos(phi1) * Math.cos(phi2)) * t_0;
	double t_3 = t_0 * t_2;
	double t_4 = t_3 + t_1;
	double tmp;
	if ((lambda1 <= -1.4) || !(lambda1 <= 3e-18)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_3 + (0.5 - (Math.cos((phi1 - phi2)) / 2.0)))), Math.sqrt((1.0 - t_4))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_4), Math.sqrt((1.0 - (t_1 + (t_2 * Math.sin((lambda2 * -0.5))))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(((math.sin((0.5 * phi1)) * math.cos((0.5 * phi2))) - (math.cos((0.5 * phi1)) * math.sin((0.5 * phi2)))), 2.0)
	t_2 = (math.cos(phi1) * math.cos(phi2)) * t_0
	t_3 = t_0 * t_2
	t_4 = t_3 + t_1
	tmp = 0
	if (lambda1 <= -1.4) or not (lambda1 <= 3e-18):
		tmp = R * (2.0 * math.atan2(math.sqrt((t_3 + (0.5 - (math.cos((phi1 - phi2)) / 2.0)))), math.sqrt((1.0 - t_4))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_4), math.sqrt((1.0 - (t_1 + (t_2 * math.sin((lambda2 * -0.5))))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0
	t_2 = Float64(Float64(cos(phi1) * cos(phi2)) * t_0)
	t_3 = Float64(t_0 * t_2)
	t_4 = Float64(t_3 + t_1)
	tmp = 0.0
	if ((lambda1 <= -1.4) || !(lambda1 <= 3e-18))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0)))), sqrt(Float64(1.0 - t_4)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_4), sqrt(Float64(1.0 - Float64(t_1 + Float64(t_2 * sin(Float64(lambda2 * -0.5)))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = ((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))) ^ 2.0;
	t_2 = (cos(phi1) * cos(phi2)) * t_0;
	t_3 = t_0 * t_2;
	t_4 = t_3 + t_1;
	tmp = 0.0;
	if ((lambda1 <= -1.4) || ~((lambda1 <= 3e-18)))
		tmp = R * (2.0 * atan2(sqrt((t_3 + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt((1.0 - t_4))));
	else
		tmp = R * (2.0 * atan2(sqrt(t_4), sqrt((1.0 - (t_1 + (t_2 * sin((lambda2 * -0.5))))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 + t$95$1), $MachinePrecision]}, If[Or[LessEqual[lambda1, -1.4], N[Not[LessEqual[lambda1, 3e-18]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$4), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$4], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 + N[(t$95$2 * N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.3999999999999999 or 2.99999999999999983e-18 < lambda1

   1. Initial program 50.1%

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

      1. div-sub 50.1%

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

      2. sin-diff 51.6%

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

      3. div-inv 51.6%

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

      4. metadata-eval 51.6%

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

      5. div-inv 51.6%

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

      6. metadata-eval 51.6%

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

      7. div-inv 51.6%

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

      8. metadata-eval 51.6%

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

      9. div-inv 51.6%

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

      10. metadata-eval 51.6%

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

   4. Applied egg-rr 51.6%

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

      1. unpow2 38.2%

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

      2. sin-mult 38.2%

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

      3. div-inv 38.2%

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

      4. metadata-eval 38.2%

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

      5. div-inv 38.2%

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

      6. metadata-eval 38.2%

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

      7. div-inv 38.2%

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

      8. metadata-eval 38.2%

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

      9. div-inv 38.2%

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

      10. metadata-eval 38.2%

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

   6. Applied egg-rr 51.6%

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

      1. div-sub 38.2%

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

      2. +-inverses 38.2%

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

      3. cos-0 38.2%

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

      4. metadata-eval 38.2%

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

      5. distribute-lft-out 38.2%

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

      6. metadata-eval 38.2%

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

      7. *-rgt-identity 38.2%

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

   8. Simplified 51.6%

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

   if -1.3999999999999999 < lambda1 < 2.99999999999999983e-18

   1. Initial program 82.1%

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

      1. div-sub 82.1%

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

      2. sin-diff 82.9%

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

      3. div-inv 82.9%

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

      4. metadata-eval 82.9%

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

      5. div-inv 82.9%

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

      6. metadata-eval 82.9%

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

      7. div-inv 82.9%

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

      8. metadata-eval 82.9%

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

      9. div-inv 82.9%

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

      10. metadata-eval 82.9%

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

   4. Applied egg-rr 82.9%

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

   5. Taylor expanded in lambda1 around 0 82.9%

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

      1. div-sub 82.1%

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

      2. sin-diff 82.9%

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

      3. div-inv 82.9%

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

      4. metadata-eval 82.9%

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

      5. div-inv 82.9%

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

      6. metadata-eval 82.9%

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

      7. div-inv 82.9%

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

      8. metadata-eval 82.9%

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

      9. div-inv 82.9%

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

      10. metadata-eval 82.9%

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

   7. Applied egg-rr 97.6%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.4 \lor \neg \left(\lambda_1 \leq 3 \cdot 10^{-18}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{1 - \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2}}}{\sqrt{1 - \left({\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.7× speedup

Math

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

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
           (-
            (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
            (* (cos (* 0.5 phi1)) (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(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * 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)) + (((sin((0.5d0 * phi1)) * cos((0.5d0 * phi2))) - (cos((0.5d0 * phi1)) * 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.sin((0.5 * phi1)) * Math.cos((0.5 * phi2))) - (Math.cos((0.5 * phi1)) * 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.sin((0.5 * phi1)) * math.cos((0.5 * phi2))) - (math.cos((0.5 * phi1)) * 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(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(0.5 * phi1)) * 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)) + (((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * 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[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $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 65.5%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

4. Applied egg-rr 66.6%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

6. Applied egg-rr 78.0%

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

7. Final simplification 78.0%

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

Alternative 6: 63.0% accurate, 0.8× speedup

Math

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

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
         (pow
          (-
           (* (sin (* 0.5 phi1)) (cos (* 0.5 phi2)))
           (* (cos (* 0.5 phi1)) (sin (* 0.5 phi2))))
          2.0))
        t_1)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = (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 - pow(((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))), 2.0)) - 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 = (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 - (((sin((0.5d0 * phi1)) * cos((0.5d0 * phi2))) - (cos((0.5d0 * phi1)) * sin((0.5d0 * phi2)))) ** 2.0d0)) - 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.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.pow(((Math.sin((0.5 * phi1)) * Math.cos((0.5 * phi2))) - (Math.cos((0.5 * phi1)) * Math.sin((0.5 * phi2)))), 2.0)) - 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.pow(((math.sin((0.5 * phi1)) * math.cos((0.5 * phi2))) - (math.cos((0.5 * phi1)) * math.sin((0.5 * phi2)))), 2.0)) - 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(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0)) - 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 - (((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))) ^ 2.0)) - 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[(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[Power[N[(N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 65.5%

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

   1. associate-*l* 65.5%

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

3. Simplified 65.5%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

6. Applied egg-rr 66.6%

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

7. Final simplification 66.6%

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

Alternative 7: 63.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

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.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((1.0 - (t_1 + math.pow(((math.sin((0.5 * phi1)) * math.cos((0.5 * phi2))) - (math.cos((0.5 * phi1)) * math.sin((0.5 * phi2)))), 2.0))))))

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 + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(t_1 + (Float64(Float64(sin(Float64(0.5 * phi1)) * cos(Float64(0.5 * phi2))) - Float64(cos(Float64(0.5 * phi1)) * sin(Float64(0.5 * phi2)))) ^ 2.0)))))))
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 + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((1.0 - (t_1 + (((sin((0.5 * phi1)) * cos((0.5 * phi2))) - (cos((0.5 * phi1)) * sin((0.5 * phi2)))) ^ 2.0))))));
end

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. div-sub 65.5%

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

   2. sin-diff 66.6%

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

   3. div-inv 66.6%

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

   4. metadata-eval 66.6%

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

   5. div-inv 66.6%

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

   6. metadata-eval 66.6%

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

   7. div-inv 66.6%

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

   8. metadata-eval 66.6%

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

   9. div-inv 66.6%

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

   10. metadata-eval 66.6%

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

4. Applied egg-rr 66.6%

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

5. Final simplification 66.6%

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

Alternative 8: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;t\_0 \leq -0.0001 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= t_0 -0.0001) (not (<= t_0 5e-12)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (* 0.5 phi1)) 2.0)
          (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (sqrt
         (-
          (pow (cos (* 0.5 phi1)) 2.0)
          (* (cos phi1) (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((t_0 <= -0.0001) || !(t_0 <= 5e-12)) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((0.5 * phi1)), 2.0) + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))), sqrt((pow(cos((0.5 * phi1)), 2.0) - (cos(phi1) * pow(sin((-0.5 * (lambda2 - lambda1))), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    if ((t_0 <= (-0.0001d0)) .or. (.not. (t_0 <= 5d-12))) then
        tmp = r * (2.0d0 * atan2(sqrt(((sin((0.5d0 * phi1)) ** 2.0d0) + (cos(phi1) * (sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0)))), sqrt(((cos((0.5d0 * phi1)) ** 2.0d0) - (cos(phi1) * (sin(((-0.5d0) * (lambda2 - lambda1))) ** 2.0d0))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((t_0 <= -0.0001) || !(t_0 <= 5e-12)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((0.5 * phi1)), 2.0) + (Math.cos(phi1) * Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0)))), Math.sqrt((Math.pow(Math.cos((0.5 * phi1)), 2.0) - (Math.cos(phi1) * Math.pow(Math.sin((-0.5 * (lambda2 - lambda1))), 2.0))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0)) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if (t_0 <= -0.0001) or not (t_0 <= 5e-12):
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((0.5 * phi1)), 2.0) + (math.cos(phi1) * math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)))), math.sqrt((math.pow(math.cos((0.5 * phi1)), 2.0) - (math.cos(phi1) * math.pow(math.sin((-0.5 * (lambda2 - lambda1))), 2.0))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((1.0 - math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((t_0 <= -0.0001) || !(t_0 <= 5e-12))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(0.5 * phi1)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))), sqrt(Float64((cos(Float64(0.5 * phi1)) ^ 2.0) - Float64(cos(phi1) * (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if ((t_0 <= -0.0001) || ~((t_0 <= 5e-12)))
		tmp = R * (2.0 * atan2(sqrt(((sin((0.5 * phi1)) ^ 2.0) + (cos(phi1) * (sin((0.5 * (lambda1 - lambda2))) ^ 2.0)))), sqrt(((cos((0.5 * phi1)) ^ 2.0) - (cos(phi1) * (sin((-0.5 * (lambda2 - lambda1))) ^ 2.0))))));
	else
		tmp = R * (2.0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((1.0 - (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.0001], N[Not[LessEqual[t$95$0, 5e-12]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[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 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) < -1.00000000000000005e-4 or 4.9999999999999997e-12 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))

   1. Initial program 59.7%

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

   3. Taylor expanded in phi2 around 0 46.9%

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

   4. Taylor expanded in phi2 around 0 47.6%

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

      1. +-commutative 7.6%

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

      2. associate--r+ 7.6%

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

      3. unpow2 7.6%

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

      4. 1-sub-sin 7.6%

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

      5. unpow2 7.6%

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

      6. sub-neg 7.6%

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

      7. mul-1-neg 7.6%

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

      8. distribute-lft-in 7.6%

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

      9. metadata-eval 7.6%

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

      10. associate-*r* 7.6%

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

      11. associate-*r* 7.6%

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

      12. metadata-eval 7.6%

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

      13. distribute-lft-in 7.6%

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

      14. +-commutative 7.6%

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

   6. Simplified 47.6%

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

   7. Taylor expanded in phi2 around 0 48.6%

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

   if -1.00000000000000005e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) < 4.9999999999999997e-12

   1. Initial program 83.3%

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

   3. Taylor expanded in lambda1 around 0 83.3%

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

   4. Taylor expanded in lambda2 around 0 82.6%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -0.0001 \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 62.2% accurate, 1.1× speedup

Math

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

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 (sin (/ (- phi1 phi2) 2.0)) 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(sin(((phi1 - phi2) / 2.0)), 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 + (sin(((phi1 - phi2) / 2.0d0)) ** 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.sin(((phi1 - phi2) / 2.0)), 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.sin(((phi1 - phi2) / 2.0)), 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 + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 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 + (sin(((phi1 - phi2) / 2.0)) ^ 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[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $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 65.5%

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

   1. unpow2 51.6%

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

   2. sin-mult 49.4%

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

   3. div-inv 49.4%

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

   4. metadata-eval 49.4%

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

   5. div-inv 49.4%

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

   6. metadata-eval 49.4%

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

   7. div-inv 49.4%

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

   8. metadata-eval 49.4%

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

   9. div-inv 49.4%

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

   10. metadata-eval 49.4%

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

4. Applied egg-rr 65.6%

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

   1. div-sub 49.4%

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

   2. +-inverses 49.4%

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

   3. cos-0 49.4%

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

   4. metadata-eval 49.4%

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

   5. distribute-lft-out 49.4%

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

   6. metadata-eval 49.4%

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

   7. *-rgt-identity 49.4%

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

6. Simplified 65.6%

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

7. Final simplification 65.6%

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

Alternative 10: 62.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_3 := \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t\_0}\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_1 \leq -0.056:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_1, t\_2\right)}}{t\_3}\right)\\ \mathbf{elif}\;\phi_1 \leq 5.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_4\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \cos \phi_1 \cdot t\_1}}{t\_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (pow (sin (* 0.5 phi1)) 2.0))
        (t_3 (sqrt (- (pow (cos (* 0.5 phi1)) 2.0) (* (cos phi1) t_0))))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= phi1 -0.056)
     (* R (* 2.0 (atan2 (sqrt (fma (cos phi1) t_1 t_2)) t_3)))
     (if (<= phi1 5.1e-6)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (* t_4 (* (* (cos phi1) (cos phi2)) t_4))
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_0))))))
       (* R (* 2.0 (atan2 (sqrt (+ t_2 (* (cos phi1) t_1))) t_3)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((-0.5 * (lambda2 - lambda1))), 2.0);
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = pow(sin((0.5 * phi1)), 2.0);
	double t_3 = sqrt((pow(cos((0.5 * phi1)), 2.0) - (cos(phi1) * t_0)));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (phi1 <= -0.056) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_1, t_2)), t_3));
	} else if (phi1 <= 5.1e-6) {
		tmp = R * (2.0 * atan2(sqrt(((t_4 * ((cos(phi1) * cos(phi2)) * t_4)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((pow(cos((phi2 * -0.5)), 2.0) - (cos(phi2) * t_0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (cos(phi1) * t_1))), t_3));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = sin(Float64(0.5 * phi1)) ^ 2.0
	t_3 = sqrt(Float64((cos(Float64(0.5 * phi1)) ^ 2.0) - Float64(cos(phi1) * t_0)))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (phi1 <= -0.056)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_1, t_2)), t_3)));
	elseif (phi1 <= 5.1e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_4 * Float64(Float64(cos(phi1) * cos(phi2)) * t_4)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - Float64(cos(phi2) * t_0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(cos(phi1) * t_1))), t_3)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.056], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 5.1e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$4 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$4), $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[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0560000000000000012

   1. Initial program 54.7%

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

   3. Taylor expanded in phi2 around 0 54.4%

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

   4. Taylor expanded in phi2 around 0 55.5%

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

      1. +-commutative 7.9%

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

      2. associate--r+ 7.9%

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

      3. unpow2 7.9%

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

      4. 1-sub-sin 7.9%

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

      5. unpow2 7.9%

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

      6. sub-neg 7.9%

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

      7. mul-1-neg 7.9%

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

      8. distribute-lft-in 7.9%

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

      9. metadata-eval 7.9%

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

      10. associate-*r* 7.9%

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

      11. associate-*r* 7.9%

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

      12. metadata-eval 7.9%

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

      13. distribute-lft-in 7.9%

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

      14. +-commutative 7.9%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in phi2 around 0 56.6%

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

      1. fma-def 56.6%

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

   9. Simplified 56.6%

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

   if -0.0560000000000000012 < phi1 < 5.1000000000000003e-6

   1. Initial program 78.7%

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

   3. Taylor expanded in phi1 around 0 78.7%

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

   5. Simplified 78.8%

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

   if 5.1000000000000003e-6 < phi1

   1. Initial program 47.3%

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

   3. Taylor expanded in phi2 around 0 47.6%

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

   4. Taylor expanded in phi2 around 0 48.5%

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

      1. +-commutative 7.9%

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

      2. associate--r+ 7.9%

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

      3. unpow2 7.9%

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

      4. 1-sub-sin 7.9%

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

      5. unpow2 7.9%

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

      6. sub-neg 7.9%

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

      7. mul-1-neg 7.9%

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

      8. distribute-lft-in 7.9%

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

      9. metadata-eval 7.9%

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

      10. associate-*r* 7.9%

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

      11. associate-*r* 7.9%

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

      12. metadata-eval 7.9%

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

      13. distribute-lft-in 7.9%

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

      14. +-commutative 7.9%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in phi2 around 0 49.0%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.056:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\phi_1 \leq 5.1 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 61.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ t_1 := {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}\\ t_2 := {\sin \left(0.5 \cdot \phi_1\right)}^{2}\\ t_3 := \sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t\_0}\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_1 \leq -0.056:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, t\_1, t\_2\right)}}{t\_3}\right)\\ \mathbf{elif}\;\phi_1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_4 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_4\right) + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t\_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_2 + \cos \phi_1 \cdot t\_1}}{t\_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))
        (t_1 (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))
        (t_2 (pow (sin (* 0.5 phi1)) 2.0))
        (t_3 (sqrt (- (pow (cos (* 0.5 phi1)) 2.0) (* (cos phi1) t_0))))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= phi1 -0.056)
     (* R (* 2.0 (atan2 (sqrt (fma (cos phi1) t_1 t_2)) t_3)))
     (if (<= phi1 8.2e-7)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (* t_4 (* (* (cos phi1) (cos phi2)) t_4))
            (pow (sin (* phi2 -0.5)) 2.0)))
          (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_0))))))
       (* R (* 2.0 (atan2 (sqrt (+ t_2 (* (cos phi1) t_1))) t_3)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((-0.5 * (lambda2 - lambda1))), 2.0);
	double t_1 = pow(sin((0.5 * (lambda1 - lambda2))), 2.0);
	double t_2 = pow(sin((0.5 * phi1)), 2.0);
	double t_3 = sqrt((pow(cos((0.5 * phi1)), 2.0) - (cos(phi1) * t_0)));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (phi1 <= -0.056) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), t_1, t_2)), t_3));
	} else if (phi1 <= 8.2e-7) {
		tmp = R * (2.0 * atan2(sqrt(((t_4 * ((cos(phi1) * cos(phi2)) * t_4)) + pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((phi2 * -0.5)), 2.0) - (cos(phi2) * t_0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (cos(phi1) * t_1))), t_3));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0
	t_1 = sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0
	t_2 = sin(Float64(0.5 * phi1)) ^ 2.0
	t_3 = sqrt(Float64((cos(Float64(0.5 * phi1)) ^ 2.0) - Float64(cos(phi1) * t_0)))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (phi1 <= -0.056)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), t_1, t_2)), t_3)));
	elseif (phi1 <= 8.2e-7)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_4 * Float64(Float64(cos(phi1) * cos(phi2)) * t_4)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - Float64(cos(phi2) * t_0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(cos(phi1) * t_1))), t_3)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -0.056], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * t$95$1 + t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 8.2e-7], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$4 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$4), $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[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.0560000000000000012

   1. Initial program 54.7%

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

   3. Taylor expanded in phi2 around 0 54.4%

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

   4. Taylor expanded in phi2 around 0 55.5%

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

      1. +-commutative 7.9%

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

      2. associate--r+ 7.9%

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

      3. unpow2 7.9%

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

      4. 1-sub-sin 7.9%

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

      5. unpow2 7.9%

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

      6. sub-neg 7.9%

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

      7. mul-1-neg 7.9%

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

      8. distribute-lft-in 7.9%

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

      9. metadata-eval 7.9%

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

      10. associate-*r* 7.9%

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

      11. associate-*r* 7.9%

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

      12. metadata-eval 7.9%

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

      13. distribute-lft-in 7.9%

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

      14. +-commutative 7.9%

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

   6. Simplified 55.6%

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

   7. Taylor expanded in phi2 around 0 56.6%

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

      1. fma-def 56.6%

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

   9. Simplified 56.6%

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

   if -0.0560000000000000012 < phi1 < 8.1999999999999998e-7

   1. Initial program 78.7%

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

   3. Taylor expanded in phi1 around 0 78.7%

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

   5. Simplified 78.8%

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

   6. Taylor expanded in phi1 around 0 77.2%

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

   if 8.1999999999999998e-7 < phi1

   1. Initial program 47.3%

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

   3. Taylor expanded in phi2 around 0 47.6%

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

   4. Taylor expanded in phi2 around 0 48.5%

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

      1. +-commutative 7.9%

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

      2. associate--r+ 7.9%

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

      3. unpow2 7.9%

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

      4. 1-sub-sin 7.9%

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

      5. unpow2 7.9%

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

      6. sub-neg 7.9%

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

      7. mul-1-neg 7.9%

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

      8. distribute-lft-in 7.9%

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

      9. metadata-eval 7.9%

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

      10. associate-*r* 7.9%

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

      11. associate-*r* 7.9%

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

      12. metadata-eval 7.9%

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

      13. distribute-lft-in 7.9%

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

      14. +-commutative 7.9%

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

   6. Simplified 48.6%

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

   7. Taylor expanded in phi2 around 0 49.0%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.056:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{elif}\;\phi_1 \leq 8.2 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -38000000000 \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0)))
   (if (or (<= phi2 -38000000000.0) (not (<= phi2 7.5e-5)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
          (- 0.5 (/ (cos (- phi1 phi2)) 2.0))))
        (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_1))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi1)
          (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0)
          (pow (sin (* 0.5 phi1)) 2.0)))
        (sqrt (- (pow (cos (* 0.5 phi1)) 2.0) (* (cos phi1) 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((-0.5 * (lambda2 - lambda1))), 2.0);
	double tmp;
	if ((phi2 <= -38000000000.0) || !(phi2 <= 7.5e-5)) {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt((pow(cos((phi2 * -0.5)), 2.0) - (cos(phi2) * t_1)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), pow(sin((0.5 * (lambda1 - lambda2))), 2.0), pow(sin((0.5 * phi1)), 2.0))), sqrt((pow(cos((0.5 * phi1)), 2.0) - (cos(phi1) * t_1)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -38000000000.0) || !(phi2 <= 7.5e-5))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0)))), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - Float64(cos(phi2) * t_1))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0), (sin(Float64(0.5 * phi1)) ^ 2.0))), sqrt(Float64((cos(Float64(0.5 * phi1)) ^ 2.0) - Float64(cos(phi1) * t_1))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -38000000000.0], N[Not[LessEqual[phi2, 7.5e-5]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.8e10 or 7.49999999999999934e-5 < phi2

   1. Initial program 51.2%

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

   3. Taylor expanded in phi1 around 0 52.3%

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

   5. Simplified 52.4%

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

      1. unpow2 52.4%

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

      2. sin-mult 52.4%

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

      3. div-inv 52.4%

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

      4. metadata-eval 52.4%

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

      5. div-inv 52.4%

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

      6. metadata-eval 52.4%

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

      7. div-inv 52.4%

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

      8. metadata-eval 52.4%

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

      9. div-inv 52.4%

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

      10. metadata-eval 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. div-sub 52.4%

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

      2. +-inverses 52.4%

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

      3. cos-0 52.4%

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

      4. metadata-eval 52.4%

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

      5. distribute-lft-out 52.4%

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

      6. metadata-eval 52.4%

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

      7. *-rgt-identity 52.4%

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

   9. Simplified 52.4%

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

   if -3.8e10 < phi2 < 7.49999999999999934e-5

   1. Initial program 77.9%

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

   3. Taylor expanded in phi2 around 0 78.1%

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

   4. Taylor expanded in phi2 around 0 78.1%

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

      1. +-commutative 7.6%

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

      2. associate--r+ 7.6%

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

      3. unpow2 7.6%

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

      4. 1-sub-sin 7.6%

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

      5. unpow2 7.6%

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

      6. sub-neg 7.6%

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

      7. mul-1-neg 7.6%

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

      8. distribute-lft-in 7.6%

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

      9. metadata-eval 7.6%

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

      10. associate-*r* 7.6%

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

      11. associate-*r* 7.6%

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

      12. metadata-eval 7.6%

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

      13. distribute-lft-in 7.6%

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

      14. +-commutative 7.6%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in phi2 around 0 75.3%

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

      1. fma-def 75.3%

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

   9. Simplified 75.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -38000000000 \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}, {\sin \left(0.5 \cdot \phi_1\right)}^{2}\right)}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 43.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;t\_0 \leq -0.0001 \lor \neg \left(t\_0 \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_1}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (+
           (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))))
   (if (or (<= t_0 -0.0001) (not (<= t_0 5e-12)))
     (*
      R
      (*
       2.0
       (atan2
        t_1
        (sqrt (- 1.0 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))))))
     (*
      R
      (*
       2.0
       (atan2 t_1 (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + pow(sin(((phi1 - phi2) / 2.0)), 2.0)));
	double tmp;
	if ((t_0 <= -0.0001) || !(t_0 <= 5e-12)) {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - pow(sin((-0.5 * (lambda2 - lambda1))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0)))
    if ((t_0 <= (-0.0001d0)) .or. (.not. (t_0 <= 5d-12))) then
        tmp = r * (2.0d0 * atan2(t_1, sqrt((1.0d0 - (sin(((-0.5d0) * (lambda2 - lambda1))) ** 2.0d0)))))
    else
        tmp = r * (2.0d0 * atan2(t_1, sqrt((1.0d0 - (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.sqrt(((t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0)) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0)));
	double tmp;
	if ((t_0 <= -0.0001) || !(t_0 <= 5e-12)) {
		tmp = R * (2.0 * Math.atan2(t_1, Math.sqrt((1.0 - Math.pow(Math.sin((-0.5 * (lambda2 - lambda1))), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_1, Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.sqrt(((t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)))
	tmp = 0
	if (t_0 <= -0.0001) or not (t_0 <= 5e-12):
		tmp = R * (2.0 * math.atan2(t_1, math.sqrt((1.0 - math.pow(math.sin((-0.5 * (lambda2 - lambda1))), 2.0)))))
	else:
		tmp = R * (2.0 * math.atan2(t_1, math.sqrt((1.0 - math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sqrt(Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)))
	tmp = 0.0
	if ((t_0 <= -0.0001) || !(t_0 <= 5e-12))
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0)));
	tmp = 0.0;
	if ((t_0 <= -0.0001) || ~((t_0 <= 5e-12)))
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - (sin((-0.5 * (lambda2 - lambda1))) ^ 2.0)))));
	else
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.0001], N[Not[LessEqual[t$95$0, 5e-12]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - 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 (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) < -1.00000000000000005e-4 or 4.9999999999999997e-12 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2))

   1. Initial program 59.7%

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

   3. Taylor expanded in phi1 around 0 46.7%

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

   5. Simplified 46.7%

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

   6. Taylor expanded in phi2 around 0 34.8%

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

   if -1.00000000000000005e-4 < (sin.f64 (/.f64 (-.f64 lambda1 lambda2) 2)) < 4.9999999999999997e-12

   1. Initial program 83.3%

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

   3. Taylor expanded in lambda1 around 0 83.3%

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

   4. Taylor expanded in lambda2 around 0 82.6%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq -0.0001 \lor \neg \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \leq 5 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 60.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}\\ \mathbf{if}\;\phi_2 \leq -38000000000 \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t\_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t\_0\right) + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t\_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot t\_1}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0)))
   (if (or (<= phi2 -38000000000.0) (not (<= phi2 7.5e-5)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_0 (* (* (cos phi1) (cos phi2)) t_0))
          (- 0.5 (/ (cos (- phi1 phi2)) 2.0))))
        (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_1))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (* 0.5 phi1)) 2.0)
          (* (cos phi1) (pow (sin (* 0.5 (- lambda1 lambda2))) 2.0))))
        (sqrt (- (pow (cos (* 0.5 phi1)) 2.0) (* (cos phi1) 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((-0.5 * (lambda2 - lambda1))), 2.0);
	double tmp;
	if ((phi2 <= -38000000000.0) || !(phi2 <= 7.5e-5)) {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt((pow(cos((phi2 * -0.5)), 2.0) - (cos(phi2) * t_1)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((0.5 * phi1)), 2.0) + (cos(phi1) * pow(sin((0.5 * (lambda1 - lambda2))), 2.0)))), sqrt((pow(cos((0.5 * phi1)), 2.0) - (cos(phi1) * t_1)))));
	}
	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) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sin(((-0.5d0) * (lambda2 - lambda1))) ** 2.0d0
    if ((phi2 <= (-38000000000.0d0)) .or. (.not. (phi2 <= 7.5d-5))) then
        tmp = r * (2.0d0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (0.5d0 - (cos((phi1 - phi2)) / 2.0d0)))), sqrt(((cos((phi2 * (-0.5d0))) ** 2.0d0) - (cos(phi2) * t_1)))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin((0.5d0 * phi1)) ** 2.0d0) + (cos(phi1) * (sin((0.5d0 * (lambda1 - lambda2))) ** 2.0d0)))), sqrt(((cos((0.5d0 * phi1)) ** 2.0d0) - (cos(phi1) * t_1)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin((-0.5 * (lambda2 - lambda1))), 2.0);
	double tmp;
	if ((phi2 <= -38000000000.0) || !(phi2 <= 7.5e-5)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0)) + (0.5 - (Math.cos((phi1 - phi2)) / 2.0)))), Math.sqrt((Math.pow(Math.cos((phi2 * -0.5)), 2.0) - (Math.cos(phi2) * t_1)))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((0.5 * phi1)), 2.0) + (Math.cos(phi1) * Math.pow(Math.sin((0.5 * (lambda1 - lambda2))), 2.0)))), Math.sqrt((Math.pow(Math.cos((0.5 * phi1)), 2.0) - (Math.cos(phi1) * t_1)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin((-0.5 * (lambda2 - lambda1))), 2.0)
	tmp = 0
	if (phi2 <= -38000000000.0) or not (phi2 <= 7.5e-5):
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)) + (0.5 - (math.cos((phi1 - phi2)) / 2.0)))), math.sqrt((math.pow(math.cos((phi2 * -0.5)), 2.0) - (math.cos(phi2) * t_1)))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((0.5 * phi1)), 2.0) + (math.cos(phi1) * math.pow(math.sin((0.5 * (lambda1 - lambda2))), 2.0)))), math.sqrt((math.pow(math.cos((0.5 * phi1)), 2.0) - (math.cos(phi1) * t_1)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0
	tmp = 0.0
	if ((phi2 <= -38000000000.0) || !(phi2 <= 7.5e-5))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0)) + Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0)))), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - Float64(cos(phi2) * t_1))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(0.5 * phi1)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(0.5 * Float64(lambda1 - lambda2))) ^ 2.0)))), sqrt(Float64((cos(Float64(0.5 * phi1)) ^ 2.0) - Float64(cos(phi1) * t_1))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = sin((-0.5 * (lambda2 - lambda1))) ^ 2.0;
	tmp = 0.0;
	if ((phi2 <= -38000000000.0) || ~((phi2 <= 7.5e-5)))
		tmp = R * (2.0 * atan2(sqrt(((t_0 * ((cos(phi1) * cos(phi2)) * t_0)) + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt(((cos((phi2 * -0.5)) ^ 2.0) - (cos(phi2) * t_1)))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin((0.5 * phi1)) ^ 2.0) + (cos(phi1) * (sin((0.5 * (lambda1 - lambda2))) ^ 2.0)))), sqrt(((cos((0.5 * phi1)) ^ 2.0) - (cos(phi1) * t_1)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi2, -38000000000.0], N[Not[LessEqual[phi2, 7.5e-5]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(0.5 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -3.8e10 or 7.49999999999999934e-5 < phi2

   1. Initial program 51.2%

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

   3. Taylor expanded in phi1 around 0 52.3%

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

   5. Simplified 52.4%

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

      1. unpow2 52.4%

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

      2. sin-mult 52.4%

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

      3. div-inv 52.4%

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

      4. metadata-eval 52.4%

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

      5. div-inv 52.4%

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

      6. metadata-eval 52.4%

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

      7. div-inv 52.4%

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

      8. metadata-eval 52.4%

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

      9. div-inv 52.4%

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

      10. metadata-eval 52.4%

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

   7. Applied egg-rr 52.4%

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

      1. div-sub 52.4%

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

      2. +-inverses 52.4%

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

      3. cos-0 52.4%

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

      4. metadata-eval 52.4%

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

      5. distribute-lft-out 52.4%

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

      6. metadata-eval 52.4%

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

      7. *-rgt-identity 52.4%

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

   9. Simplified 52.4%

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

   if -3.8e10 < phi2 < 7.49999999999999934e-5

   1. Initial program 77.9%

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

   3. Taylor expanded in phi2 around 0 78.1%

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

   4. Taylor expanded in phi2 around 0 78.1%

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

      1. +-commutative 7.6%

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

      2. associate--r+ 7.6%

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

      3. unpow2 7.6%

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

      4. 1-sub-sin 7.6%

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

      5. unpow2 7.6%

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

      6. sub-neg 7.6%

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

      7. mul-1-neg 7.6%

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

      8. distribute-lft-in 7.6%

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

      9. metadata-eval 7.6%

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

      10. associate-*r* 7.6%

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

      11. associate-*r* 7.6%

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

      12. metadata-eval 7.6%

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

      13. distribute-lft-in 7.6%

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

      14. +-commutative 7.6%

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

   6. Simplified 78.1%

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

   7. Taylor expanded in phi2 around 0 75.3%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -38000000000 \lor \neg \left(\phi_2 \leq 7.5 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \phi_1\right)}^{2} + \cos \phi_1 \cdot {\sin \left(0.5 \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{2}}}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 34.7% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in phi1 around 0 51.5%

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

5. Simplified 51.6%

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

6. Taylor expanded in phi2 around 0 36.8%

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

7. Final simplification 36.8%

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

Alternative 16: 15.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \phi_2 \cdot t\_0\\ t_2 := \sqrt{{t\_0}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-133}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \phi_1\right) + -0.5 \cdot t\_1}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{0.5 \cdot t\_1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1 (* phi2 t_0))
        (t_2
         (sqrt
          (-
           (pow t_0 2.0)
           (* (cos phi1) (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))))))
   (if (<= phi2 6.8e-133)
     (* R (* 2.0 (atan2 (+ (sin (* 0.5 phi1)) (* -0.5 t_1)) t_2)))
     (* R (* 2.0 (atan2 (* 0.5 t_1) t_2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = phi2 * t_0;
	double t_2 = sqrt((pow(t_0, 2.0) - (cos(phi1) * pow(sin((-0.5 * (lambda2 - lambda1))), 2.0))));
	double tmp;
	if (phi2 <= 6.8e-133) {
		tmp = R * (2.0 * atan2((sin((0.5 * phi1)) + (-0.5 * t_1)), t_2));
	} else {
		tmp = R * (2.0 * atan2((0.5 * t_1), t_2));
	}
	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 = cos((0.5d0 * phi1))
    t_1 = phi2 * t_0
    t_2 = sqrt(((t_0 ** 2.0d0) - (cos(phi1) * (sin(((-0.5d0) * (lambda2 - lambda1))) ** 2.0d0))))
    if (phi2 <= 6.8d-133) then
        tmp = r * (2.0d0 * atan2((sin((0.5d0 * phi1)) + ((-0.5d0) * t_1)), t_2))
    else
        tmp = r * (2.0d0 * atan2((0.5d0 * t_1), t_2))
    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.cos((0.5 * phi1));
	double t_1 = phi2 * t_0;
	double t_2 = Math.sqrt((Math.pow(t_0, 2.0) - (Math.cos(phi1) * Math.pow(Math.sin((-0.5 * (lambda2 - lambda1))), 2.0))));
	double tmp;
	if (phi2 <= 6.8e-133) {
		tmp = R * (2.0 * Math.atan2((Math.sin((0.5 * phi1)) + (-0.5 * t_1)), t_2));
	} else {
		tmp = R * (2.0 * Math.atan2((0.5 * t_1), t_2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	t_1 = phi2 * t_0
	t_2 = math.sqrt((math.pow(t_0, 2.0) - (math.cos(phi1) * math.pow(math.sin((-0.5 * (lambda2 - lambda1))), 2.0))))
	tmp = 0
	if phi2 <= 6.8e-133:
		tmp = R * (2.0 * math.atan2((math.sin((0.5 * phi1)) + (-0.5 * t_1)), t_2))
	else:
		tmp = R * (2.0 * math.atan2((0.5 * t_1), t_2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = Float64(phi2 * t_0)
	t_2 = sqrt(Float64((t_0 ^ 2.0) - Float64(cos(phi1) * (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))
	tmp = 0.0
	if (phi2 <= 6.8e-133)
		tmp = Float64(R * Float64(2.0 * atan(Float64(sin(Float64(0.5 * phi1)) + Float64(-0.5 * t_1)), t_2)));
	else
		tmp = Float64(R * Float64(2.0 * atan(Float64(0.5 * t_1), t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1));
	t_1 = phi2 * t_0;
	t_2 = sqrt(((t_0 ^ 2.0) - (cos(phi1) * (sin((-0.5 * (lambda2 - lambda1))) ^ 2.0))));
	tmp = 0.0;
	if (phi2 <= 6.8e-133)
		tmp = R * (2.0 * atan2((sin((0.5 * phi1)) + (-0.5 * t_1)), t_2));
	else
		tmp = R * (2.0 * atan2((0.5 * t_1), t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(phi2 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 6.8e-133], N[(R * N[(2.0 * N[ArcTan[N[(N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[(0.5 * t$95$1), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 6.80000000000000012e-133

   1. Initial program 68.5%

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

   3. Taylor expanded in phi2 around 0 56.6%

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

   4. Taylor expanded in phi2 around 0 57.0%

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

      1. +-commutative 10.9%

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

      2. associate--r+ 10.9%

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

      3. unpow2 10.9%

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

      4. 1-sub-sin 10.9%

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

      5. unpow2 10.9%

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

      6. sub-neg 10.9%

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

      7. mul-1-neg 10.9%

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

      8. distribute-lft-in 10.9%

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

      9. metadata-eval 10.9%

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

      10. associate-*r* 10.9%

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

      11. associate-*r* 10.9%

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

      12. metadata-eval 10.9%

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

      13. distribute-lft-in 10.9%

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

      14. +-commutative 10.9%

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

   6. Simplified 57.0%

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

   7. Taylor expanded in phi2 around -inf 16.0%

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

   if 6.80000000000000012e-133 < phi2

   1. Initial program 60.3%

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

   3. Taylor expanded in phi2 around 0 39.0%

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

   4. Taylor expanded in phi2 around 0 39.7%

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

      1. +-commutative 6.8%

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

      2. associate--r+ 6.8%

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

      3. unpow2 6.8%

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

      4. 1-sub-sin 6.8%

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

      5. unpow2 6.8%

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

      6. sub-neg 6.8%

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

      7. mul-1-neg 6.8%

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

      8. distribute-lft-in 6.8%

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

      9. metadata-eval 6.8%

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

      10. associate-*r* 6.8%

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

      11. associate-*r* 6.8%

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

      12. metadata-eval 6.8%

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

      13. distribute-lft-in 6.8%

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

      14. +-commutative 6.8%

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

   6. Simplified 39.7%

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

   7. Taylor expanded in phi2 around inf 15.2%

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

4. Final simplification 15.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 6.8 \cdot 10^{-133}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \phi_1\right) + -0.5 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{0.5 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 16.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \phi_2 \cdot t\_0\\ t_2 := \sqrt{{t\_0}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}\\ t_3 := \sin \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq -1.85 \cdot 10^{-87}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t\_3 + -0.5 \cdot t\_1}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{0.5 \cdot t\_1 - t\_3}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1 (* phi2 t_0))
        (t_2
         (sqrt
          (-
           (pow t_0 2.0)
           (* (cos phi1) (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0)))))
        (t_3 (sin (* 0.5 phi1))))
   (if (<= phi2 -1.85e-87)
     (* R (* 2.0 (atan2 (+ t_3 (* -0.5 t_1)) t_2)))
     (* R (* 2.0 (atan2 (- (* 0.5 t_1) t_3) t_2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = phi2 * t_0;
	double t_2 = sqrt((pow(t_0, 2.0) - (cos(phi1) * pow(sin((-0.5 * (lambda2 - lambda1))), 2.0))));
	double t_3 = sin((0.5 * phi1));
	double tmp;
	if (phi2 <= -1.85e-87) {
		tmp = R * (2.0 * atan2((t_3 + (-0.5 * t_1)), t_2));
	} else {
		tmp = R * (2.0 * atan2(((0.5 * t_1) - t_3), t_2));
	}
	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) :: t_3
    real(8) :: tmp
    t_0 = cos((0.5d0 * phi1))
    t_1 = phi2 * t_0
    t_2 = sqrt(((t_0 ** 2.0d0) - (cos(phi1) * (sin(((-0.5d0) * (lambda2 - lambda1))) ** 2.0d0))))
    t_3 = sin((0.5d0 * phi1))
    if (phi2 <= (-1.85d-87)) then
        tmp = r * (2.0d0 * atan2((t_3 + ((-0.5d0) * t_1)), t_2))
    else
        tmp = r * (2.0d0 * atan2(((0.5d0 * t_1) - t_3), t_2))
    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.cos((0.5 * phi1));
	double t_1 = phi2 * t_0;
	double t_2 = Math.sqrt((Math.pow(t_0, 2.0) - (Math.cos(phi1) * Math.pow(Math.sin((-0.5 * (lambda2 - lambda1))), 2.0))));
	double t_3 = Math.sin((0.5 * phi1));
	double tmp;
	if (phi2 <= -1.85e-87) {
		tmp = R * (2.0 * Math.atan2((t_3 + (-0.5 * t_1)), t_2));
	} else {
		tmp = R * (2.0 * Math.atan2(((0.5 * t_1) - t_3), t_2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	t_1 = phi2 * t_0
	t_2 = math.sqrt((math.pow(t_0, 2.0) - (math.cos(phi1) * math.pow(math.sin((-0.5 * (lambda2 - lambda1))), 2.0))))
	t_3 = math.sin((0.5 * phi1))
	tmp = 0
	if phi2 <= -1.85e-87:
		tmp = R * (2.0 * math.atan2((t_3 + (-0.5 * t_1)), t_2))
	else:
		tmp = R * (2.0 * math.atan2(((0.5 * t_1) - t_3), t_2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = Float64(phi2 * t_0)
	t_2 = sqrt(Float64((t_0 ^ 2.0) - Float64(cos(phi1) * (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))
	t_3 = sin(Float64(0.5 * phi1))
	tmp = 0.0
	if (phi2 <= -1.85e-87)
		tmp = Float64(R * Float64(2.0 * atan(Float64(t_3 + Float64(-0.5 * t_1)), t_2)));
	else
		tmp = Float64(R * Float64(2.0 * atan(Float64(Float64(0.5 * t_1) - t_3), t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1));
	t_1 = phi2 * t_0;
	t_2 = sqrt(((t_0 ^ 2.0) - (cos(phi1) * (sin((-0.5 * (lambda2 - lambda1))) ^ 2.0))));
	t_3 = sin((0.5 * phi1));
	tmp = 0.0;
	if (phi2 <= -1.85e-87)
		tmp = R * (2.0 * atan2((t_3 + (-0.5 * t_1)), t_2));
	else
		tmp = R * (2.0 * atan2(((0.5 * t_1) - t_3), t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(phi2 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -1.85e-87], N[(R * N[(2.0 * N[ArcTan[N[(t$95$3 + N[(-0.5 * t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[(N[(0.5 * t$95$1), $MachinePrecision] - t$95$3), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.8500000000000001e-87

   1. Initial program 54.6%

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

   3. Taylor expanded in phi2 around 0 29.6%

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

   4. Taylor expanded in phi2 around 0 30.3%

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

      1. +-commutative 14.7%

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

      2. associate--r+ 14.7%

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

      3. unpow2 14.7%

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

      4. 1-sub-sin 14.7%

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

      5. unpow2 14.7%

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

      6. sub-neg 14.7%

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

      7. mul-1-neg 14.7%

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

      8. distribute-lft-in 14.7%

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

      9. metadata-eval 14.7%

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

      10. associate-*r* 14.7%

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

      11. associate-*r* 14.7%

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

      12. metadata-eval 14.7%

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

      13. distribute-lft-in 14.7%

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

      14. +-commutative 14.7%

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

   6. Simplified 30.3%

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

   7. Taylor expanded in phi2 around -inf 17.5%

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

   if -1.8500000000000001e-87 < phi2

   1. Initial program 70.2%

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

   3. Taylor expanded in phi2 around 0 59.1%

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

   4. Taylor expanded in phi2 around 0 59.4%

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

      1. +-commutative 7.1%

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

      2. associate--r+ 7.1%

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

      3. unpow2 7.1%

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

      4. 1-sub-sin 7.1%

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

      5. unpow2 7.1%

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

      6. sub-neg 7.1%

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

      7. mul-1-neg 7.1%

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

      8. distribute-lft-in 7.1%

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

      9. metadata-eval 7.1%

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

      10. associate-*r* 7.1%

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

      11. associate-*r* 7.1%

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

      12. metadata-eval 7.1%

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

      13. distribute-lft-in 7.1%

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

      14. +-commutative 7.1%

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

   6. Simplified 59.5%

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

   7. Taylor expanded in phi2 around inf 18.4%

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

      1. +-commutative 18.4%

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

      2. mul-1-neg 18.4%

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

      3. unsub-neg 18.4%

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

   9. Simplified 18.4%

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

4. Final simplification 18.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.85 \cdot 10^{-87}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sin \left(0.5 \cdot \phi_1\right) + -0.5 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{0.5 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right) - \sin \left(0.5 \cdot \phi_1\right)}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 11.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ t_1 := \phi_2 \cdot t\_0\\ t_2 := \sqrt{{t\_0}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}\\ \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{-0.5 \cdot t\_1}{t\_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{0.5 \cdot t\_1}{t\_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 phi1)))
        (t_1 (* phi2 t_0))
        (t_2
         (sqrt
          (-
           (pow t_0 2.0)
           (* (cos phi1) (pow (sin (* -0.5 (- lambda2 lambda1))) 2.0))))))
   (if (<= phi2 -2e-310)
     (* R (* 2.0 (atan2 (* -0.5 t_1) t_2)))
     (* R (* 2.0 (atan2 (* 0.5 t_1) t_2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double t_1 = phi2 * t_0;
	double t_2 = sqrt((pow(t_0, 2.0) - (cos(phi1) * pow(sin((-0.5 * (lambda2 - lambda1))), 2.0))));
	double tmp;
	if (phi2 <= -2e-310) {
		tmp = R * (2.0 * atan2((-0.5 * t_1), t_2));
	} else {
		tmp = R * (2.0 * atan2((0.5 * t_1), t_2));
	}
	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 = cos((0.5d0 * phi1))
    t_1 = phi2 * t_0
    t_2 = sqrt(((t_0 ** 2.0d0) - (cos(phi1) * (sin(((-0.5d0) * (lambda2 - lambda1))) ** 2.0d0))))
    if (phi2 <= (-2d-310)) then
        tmp = r * (2.0d0 * atan2(((-0.5d0) * t_1), t_2))
    else
        tmp = r * (2.0d0 * atan2((0.5d0 * t_1), t_2))
    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.cos((0.5 * phi1));
	double t_1 = phi2 * t_0;
	double t_2 = Math.sqrt((Math.pow(t_0, 2.0) - (Math.cos(phi1) * Math.pow(Math.sin((-0.5 * (lambda2 - lambda1))), 2.0))));
	double tmp;
	if (phi2 <= -2e-310) {
		tmp = R * (2.0 * Math.atan2((-0.5 * t_1), t_2));
	} else {
		tmp = R * (2.0 * Math.atan2((0.5 * t_1), t_2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	t_1 = phi2 * t_0
	t_2 = math.sqrt((math.pow(t_0, 2.0) - (math.cos(phi1) * math.pow(math.sin((-0.5 * (lambda2 - lambda1))), 2.0))))
	tmp = 0
	if phi2 <= -2e-310:
		tmp = R * (2.0 * math.atan2((-0.5 * t_1), t_2))
	else:
		tmp = R * (2.0 * math.atan2((0.5 * t_1), t_2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	t_1 = Float64(phi2 * t_0)
	t_2 = sqrt(Float64((t_0 ^ 2.0) - Float64(cos(phi1) * (sin(Float64(-0.5 * Float64(lambda2 - lambda1))) ^ 2.0))))
	tmp = 0.0
	if (phi2 <= -2e-310)
		tmp = Float64(R * Float64(2.0 * atan(Float64(-0.5 * t_1), t_2)));
	else
		tmp = Float64(R * Float64(2.0 * atan(Float64(0.5 * t_1), t_2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1));
	t_1 = phi2 * t_0;
	t_2 = sqrt(((t_0 ^ 2.0) - (cos(phi1) * (sin((-0.5 * (lambda2 - lambda1))) ^ 2.0))));
	tmp = 0.0;
	if (phi2 <= -2e-310)
		tmp = R * (2.0 * atan2((-0.5 * t_1), t_2));
	else
		tmp = R * (2.0 * atan2((0.5 * t_1), t_2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(phi2 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(-0.5 * N[(lambda2 - lambda1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -2e-310], N[(R * N[(2.0 * N[ArcTan[N[(-0.5 * t$95$1), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[(0.5 * t$95$1), $MachinePrecision] / t$95$2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < -1.999999999999994e-310

   1. Initial program 62.7%

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

   3. Taylor expanded in phi2 around 0 46.0%

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

   4. Taylor expanded in phi2 around -inf 11.8%

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

   5. Taylor expanded in phi2 around 0 12.2%

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

      1. +-commutative 12.2%

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

      2. associate--r+ 12.2%

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

      3. unpow2 12.2%

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

      4. 1-sub-sin 12.2%

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

      5. unpow2 12.2%

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

      6. sub-neg 12.2%

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

      7. mul-1-neg 12.2%

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

      8. distribute-lft-in 12.2%

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

      9. metadata-eval 12.2%

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

      10. associate-*r* 12.2%

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

      11. associate-*r* 12.2%

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

      12. metadata-eval 12.2%

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

      13. distribute-lft-in 12.2%

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

      14. +-commutative 12.2%

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

   7. Simplified 12.2%

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

   if -1.999999999999994e-310 < phi2

   1. Initial program 67.8%

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

   3. Taylor expanded in phi2 around 0 53.7%

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

   4. Taylor expanded in phi2 around 0 54.1%

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

      1. +-commutative 7.2%

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

      2. associate--r+ 7.2%

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

      3. unpow2 7.2%

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

      4. 1-sub-sin 7.2%

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

      5. unpow2 7.2%

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

      6. sub-neg 7.2%

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

      7. mul-1-neg 7.2%

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

      8. distribute-lft-in 7.2%

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

      9. metadata-eval 7.2%

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

      10. associate-*r* 7.2%

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

      11. associate-*r* 7.2%

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

      12. metadata-eval 7.2%

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

      13. distribute-lft-in 7.2%

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

      14. +-commutative 7.2%

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

   6. Simplified 54.2%

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

   7. Taylor expanded in phi2 around inf 13.0%

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

4. Final simplification 12.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-310}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{-0.5 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{0.5 \cdot \left(\phi_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}{\sqrt{{\cos \left(0.5 \cdot \phi_1\right)}^{2} - \cos \phi_1 \cdot {\sin \left(-0.5 \cdot \left(\lambda_2 - \lambda_1\right)\right)}^{2}}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 8.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in phi2 around 0 50.2%

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

4. Taylor expanded in phi2 around -inf 9.0%

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

5. Taylor expanded in phi2 around 0 9.4%

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

   1. +-commutative 9.4%

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

   2. associate--r+ 9.4%

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

   3. unpow2 9.4%

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

   4. 1-sub-sin 9.4%

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

   5. unpow2 9.4%

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

   6. sub-neg 9.4%

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

   7. mul-1-neg 9.4%

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

   8. distribute-lft-in 9.4%

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

   9. metadata-eval 9.4%

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

   10. associate-*r* 9.4%

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

   11. associate-*r* 9.4%

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

   12. metadata-eval 9.4%

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

   13. distribute-lft-in 9.4%

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

   14. +-commutative 9.4%

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

7. Simplified 9.4%

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

8. Final simplification 9.4%

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

Alternative 20: 8.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in phi2 around 0 50.2%

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

4. Taylor expanded in phi2 around -inf 9.0%

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

5. Taylor expanded in phi1 around 0 9.4%

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

7. Simplified 9.4%

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

8. Taylor expanded in phi2 around 0 9.4%

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

9. Final simplification 9.4%

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

Alternative 21: 8.7% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

3. Taylor expanded in phi2 around 0 50.2%

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

4. Taylor expanded in phi2 around -inf 9.0%

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

5. Taylor expanded in phi1 around 0 9.4%

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

7. Simplified 9.4%

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

8. Taylor expanded in phi2 around 0 9.4%

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

9. Final simplification 9.4%

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

Reproduce

herbie shell --seed 2024031 
(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))))))))))