Distance on a great circle

Percentage Accurate: 61.5% → 62.3%

Time: 1.8min

Alternatives: 17

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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 62.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_1 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_2 := \cos \phi_2 \cdot t_1\\ t_3 := \sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_2}\\ t_4 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -3.9:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2}}{t_3}\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_4 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_4\right)}}{\sqrt{{\cos \left(\phi_1 \cdot 0.5\right)}^{2} - \cos \phi_1 \cdot t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t_1, t_0\right)}}{t_3}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* phi2 -0.5)) 2.0))
        (t_1 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_2 (* (cos phi2) t_1))
        (t_3 (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) t_2)))
        (t_4 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= phi2 -3.9)
     (* R (* 2.0 (atan2 (sqrt (+ t_0 t_2)) t_3)))
     (if (<= phi2 2e-5)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
            (* t_4 (* (* (cos phi1) (cos phi2)) t_4))))
          (sqrt (- (pow (cos (* phi1 0.5)) 2.0) (* (cos phi1) t_1))))))
       (* R (* 2.0 (atan2 (sqrt (fma (cos phi2) t_1 t_0)) t_3)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((phi2 * -0.5)), 2.0);
	double t_1 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = cos(phi2) * t_1;
	double t_3 = sqrt((pow(cos((phi2 * -0.5)), 2.0) - t_2));
	double t_4 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (phi2 <= -3.9) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + t_2)), t_3));
	} else if (phi2 <= 2e-5) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_4 * ((cos(phi1) * cos(phi2)) * t_4)))), sqrt((pow(cos((phi1 * 0.5)), 2.0) - (cos(phi1) * t_1)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), t_1, t_0)), t_3));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi2 * -0.5)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_2 = Float64(cos(phi2) * t_1)
	t_3 = sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - t_2))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (phi2 <= -3.9)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + t_2)), t_3)));
	elseif (phi2 <= 2e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_4 * Float64(Float64(cos(phi1) * cos(phi2)) * t_4)))), sqrt(Float64((cos(Float64(phi1 * 0.5)) ^ 2.0) - Float64(cos(phi1) * t_1))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), t_1, t_0)), t_3)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, -3.9], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$4 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$1 + t$95$0), $MachinePrecision]], $MachinePrecision] / t$95$3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -3.89999999999999991

   1. Initial program 43.1%

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

   2. Taylor expanded in phi1 around 0 44.1%

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

      1. associate--r+ 44.1%

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

      2. unpow2 44.1%

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

      3. 1-sub-sin 44.2%

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

      4. unpow2 44.2%

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

   4. Simplified 44.2%

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

   5. Taylor expanded in phi1 around 0 45.5%

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

   if -3.89999999999999991 < phi2 < 2.00000000000000016e-5

   1. Initial program 78.9%

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

   2. Taylor expanded in phi2 around 0 78.8%

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

      1. associate--r+ 78.8%

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

      2. unpow2 78.8%

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

      3. 1-sub-sin 78.8%

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

      4. unpow2 78.8%

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

      5. *-commutative 78.8%

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

   4. Simplified 78.8%

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

   if 2.00000000000000016e-5 < phi2

   1. Initial program 41.3%

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

   2. Taylor expanded in phi1 around 0 43.1%

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

      1. associate--r+ 43.2%

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

      2. unpow2 43.2%

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

      3. 1-sub-sin 43.3%

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

      4. unpow2 43.3%

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

   4. Simplified 43.3%

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

   5. Taylor expanded in phi1 around 0 44.4%

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

      1. +-commutative 44.4%

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

      2. fma-def 44.4%

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

   7. Simplified 44.4%

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

4. Final simplification 62.2%

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

Alternative 2: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_2 \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot \left(t_2 \cdot t_2\right)}}{\sqrt{\left(1 - t_0\right) - t_1 \cdot \left(t_2 \cdot \sin \left(\lambda_1 \cdot 0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot \left(t_1 \cdot t_2\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + t_1 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= lambda2 5.7e-5)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* t_1 (* t_2 t_2))))
        (sqrt (- (- 1.0 t_0) (* t_1 (* t_2 (sin (* lambda1 0.5)))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* t_2 (* t_1 t_2))))
        (sqrt
         (-
          1.0
          (+
           (pow (sin (* (- phi1 phi2) 0.5)) 2.0)
           (* t_1 (pow (sin (* lambda2 -0.5)) 2.0)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = cos(phi1) * cos(phi2);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (lambda2 <= 5.7e-5) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_1 * (t_2 * t_2)))), sqrt(((1.0 - t_0) - (t_1 * (t_2 * sin((lambda1 * 0.5))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_2 * (t_1 * t_2)))), sqrt((1.0 - (pow(sin(((phi1 - phi2) * 0.5)), 2.0) + (t_1 * pow(sin((lambda2 * -0.5)), 2.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_1 = cos(phi1) * cos(phi2)
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    if (lambda2 <= 5.7d-5) then
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (t_1 * (t_2 * t_2)))), sqrt(((1.0d0 - t_0) - (t_1 * (t_2 * sin((lambda1 * 0.5d0))))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (t_2 * (t_1 * t_2)))), sqrt((1.0d0 - ((sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0) + (t_1 * (sin((lambda2 * (-0.5d0))) ** 2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if (lambda2 <= 5.7e-5) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (t_1 * (t_2 * t_2)))), Math.sqrt(((1.0 - t_0) - (t_1 * (t_2 * Math.sin((lambda1 * 0.5))))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (t_2 * (t_1 * t_2)))), Math.sqrt((1.0 - (Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0) + (t_1 * Math.pow(Math.sin((lambda2 * -0.5)), 2.0)))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_1 = math.cos(phi1) * math.cos(phi2)
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if lambda2 <= 5.7e-5:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (t_1 * (t_2 * t_2)))), math.sqrt(((1.0 - t_0) - (t_1 * (t_2 * math.sin((lambda1 * 0.5))))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (t_2 * (t_1 * t_2)))), math.sqrt((1.0 - (math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0) + (t_1 * math.pow(math.sin((lambda2 * -0.5)), 2.0)))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (lambda2 <= 5.7e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(t_1 * Float64(t_2 * t_2)))), sqrt(Float64(Float64(1.0 - t_0) - Float64(t_1 * Float64(t_2 * sin(Float64(lambda1 * 0.5)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(t_2 * Float64(t_1 * t_2)))), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0) + Float64(t_1 * (sin(Float64(lambda2 * -0.5)) ^ 2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_1 = cos(phi1) * cos(phi2);
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if (lambda2 <= 5.7e-5)
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_1 * (t_2 * t_2)))), sqrt(((1.0 - t_0) - (t_1 * (t_2 * sin((lambda1 * 0.5))))))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_2 * (t_1 * t_2)))), sqrt((1.0 - ((sin(((phi1 - phi2) * 0.5)) ^ 2.0) + (t_1 * (sin((lambda2 * -0.5)) ^ 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, 5.7e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$1 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - N[(t$95$1 * N[(t$95$2 * N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$2 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.7000000000000003e-5

   1. Initial program 64.2%

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

      1. associate-*l* 64.2%

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

   3. Simplified 64.2%

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

   4. Taylor expanded in lambda2 around 0 56.1%

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

   if 5.7000000000000003e-5 < lambda2

   1. Initial program 49.1%

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

   2. Taylor expanded in lambda1 around 0 49.0%

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

4. Final simplification 54.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.7 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\lambda_1 \cdot 0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \]

Alternative 3: 61.5% accurate, 1.0× speedup

Math

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

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)))
        (t_2 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
   (* R (* 2.0 (atan2 (sqrt (+ t_2 t_1)) (sqrt (- (- 1.0 t_2) 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);
	double t_2 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	return R * (2.0 * atan2(sqrt((t_2 + t_1)), sqrt(((1.0 - t_2) - 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
    real(8) :: t_2
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = (cos(phi1) * cos(phi2)) * (t_0 * t_0)
    t_2 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    code = r * (2.0d0 * atan2(sqrt((t_2 + t_1)), sqrt(((1.0d0 - t_2) - 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);
	double t_2 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	return R * (2.0 * Math.atan2(Math.sqrt((t_2 + t_1)), Math.sqrt(((1.0 - t_2) - 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)
	t_2 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	return R * (2.0 * math.atan2(math.sqrt((t_2 + t_1)), math.sqrt(((1.0 - t_2) - 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))
	t_2 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	return Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + t_1)), sqrt(Float64(Float64(1.0 - t_2) - 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);
	t_2 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	tmp = R * (2.0 * atan2(sqrt((t_2 + t_1)), sqrt(((1.0 - t_2) - 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]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$2), $MachinePrecision] - t$95$1), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 60.9%

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

   1. associate-*l* 60.9%

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

3. Simplified 60.9%

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

4. Final simplification 60.9%

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

Alternative 4: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_2 \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2 \cdot \left(t_3 \cdot t_2\right)}}{\sqrt{1 - \left(t_1 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_3 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{\left(1 - t_0\right) - t_3 \cdot \left(t_2 \cdot t_2\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (pow (sin (* (- phi1 phi2) 0.5)) 2.0))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* (cos phi1) (cos phi2))))
   (if (<= lambda2 4.8e-6)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* t_2 (* t_3 t_2))))
        (sqrt
         (-
          1.0
          (+
           t_1
           (* (cos phi2) (* (cos phi1) (pow (sin (* lambda1 0.5)) 2.0)))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_1 (* t_3 (pow (sin (* lambda2 -0.5)) 2.0))))
        (sqrt (- (- 1.0 t_0) (* t_3 (* t_2 t_2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = pow(sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = cos(phi1) * cos(phi2);
	double tmp;
	if (lambda2 <= 4.8e-6) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_2 * (t_3 * t_2)))), sqrt((1.0 - (t_1 + (cos(phi2) * (cos(phi1) * pow(sin((lambda1 * 0.5)), 2.0))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_1 + (t_3 * pow(sin((lambda2 * -0.5)), 2.0)))), sqrt(((1.0 - t_0) - (t_3 * (t_2 * 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 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_1 = sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    t_3 = cos(phi1) * cos(phi2)
    if (lambda2 <= 4.8d-6) then
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (t_2 * (t_3 * t_2)))), sqrt((1.0d0 - (t_1 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5d0)) ** 2.0d0))))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_1 + (t_3 * (sin((lambda2 * (-0.5d0))) ** 2.0d0)))), sqrt(((1.0d0 - t_0) - (t_3 * (t_2 * 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.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_3 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (lambda2 <= 4.8e-6) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (t_2 * (t_3 * t_2)))), Math.sqrt((1.0 - (t_1 + (Math.cos(phi2) * (Math.cos(phi1) * Math.pow(Math.sin((lambda1 * 0.5)), 2.0))))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_1 + (t_3 * Math.pow(Math.sin((lambda2 * -0.5)), 2.0)))), Math.sqrt(((1.0 - t_0) - (t_3 * (t_2 * t_2))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_1 = math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	t_3 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if lambda2 <= 4.8e-6:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (t_2 * (t_3 * t_2)))), math.sqrt((1.0 - (t_1 + (math.cos(phi2) * (math.cos(phi1) * math.pow(math.sin((lambda1 * 0.5)), 2.0))))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_1 + (t_3 * math.pow(math.sin((lambda2 * -0.5)), 2.0)))), math.sqrt(((1.0 - t_0) - (t_3 * (t_2 * t_2))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda2 <= 4.8e-6)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(t_2 * Float64(t_3 * t_2)))), sqrt(Float64(1.0 - Float64(t_1 + Float64(cos(phi2) * Float64(cos(phi1) * (sin(Float64(lambda1 * 0.5)) ^ 2.0)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + Float64(t_3 * (sin(Float64(lambda2 * -0.5)) ^ 2.0)))), sqrt(Float64(Float64(1.0 - t_0) - Float64(t_3 * Float64(t_2 * t_2)))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_1 = sin(((phi1 - phi2) * 0.5)) ^ 2.0;
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	t_3 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (lambda2 <= 4.8e-6)
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_2 * (t_3 * t_2)))), sqrt((1.0 - (t_1 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5)) ^ 2.0))))))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_1 + (t_3 * (sin((lambda2 * -0.5)) ^ 2.0)))), sqrt(((1.0 - t_0) - (t_3 * (t_2 * t_2))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda2, 4.8e-6], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$2 * N[(t$95$3 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$1 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[(t$95$3 * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[(1.0 - t$95$0), $MachinePrecision] - N[(t$95$3 * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 4.7999999999999998e-6

   1. Initial program 64.2%

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

   2. Taylor expanded in lambda2 around 0 56.5%

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

   if 4.7999999999999998e-6 < lambda2

   1. Initial program 49.1%

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

      1. associate-*l* 49.1%

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

   3. Simplified 49.0%

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

   4. Taylor expanded in lambda1 around 0 49.3%

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

4. Final simplification 54.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.8 \cdot 10^{-6}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - \left({\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{\left(1 - {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\right) - \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \left(\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}\right)\\ \end{array} \]

Alternative 5: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ t_3 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(t_2 \cdot t_1\right)}\\ \mathbf{if}\;\lambda_2 \leq 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{1 - \left(t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{1 - \left(t_0 + t_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (cos phi1) (cos phi2)))
        (t_3
         (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* t_1 (* t_2 t_1))))))
   (if (<= lambda2 1e-5)
     (*
      R
      (*
       2.0
       (atan2
        t_3
        (sqrt
         (-
          1.0
          (+
           t_0
           (* (cos phi2) (* (cos phi1) (pow (sin (* lambda1 0.5)) 2.0)))))))))
     (*
      R
      (*
       2.0
       (atan2
        t_3
        (sqrt (- 1.0 (+ t_0 (* t_2 (pow (sin (* lambda2 -0.5)) 2.0)))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos(phi1) * cos(phi2);
	double t_3 = sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_2 * t_1))));
	double tmp;
	if (lambda2 <= 1e-5) {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - (t_0 + (cos(phi2) * (cos(phi1) * pow(sin((lambda1 * 0.5)), 2.0))))))));
	} else {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - (t_0 + (t_2 * pow(sin((lambda2 * -0.5)), 2.0)))))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = cos(phi1) * cos(phi2)
    t_3 = sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_1 * (t_2 * t_1))))
    if (lambda2 <= 1d-5) then
        tmp = r * (2.0d0 * atan2(t_3, sqrt((1.0d0 - (t_0 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5d0)) ** 2.0d0))))))))
    else
        tmp = r * (2.0d0 * atan2(t_3, sqrt((1.0d0 - (t_0 + (t_2 * (sin((lambda2 * (-0.5d0))) ** 2.0d0)))))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.cos(phi1) * Math.cos(phi2);
	double t_3 = Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_2 * t_1))));
	double tmp;
	if (lambda2 <= 1e-5) {
		tmp = R * (2.0 * Math.atan2(t_3, Math.sqrt((1.0 - (t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.pow(Math.sin((lambda1 * 0.5)), 2.0))))))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_3, Math.sqrt((1.0 - (t_0 + (t_2 * Math.pow(Math.sin((lambda2 * -0.5)), 2.0)))))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.cos(phi1) * math.cos(phi2)
	t_3 = math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_2 * t_1))))
	tmp = 0
	if lambda2 <= 1e-5:
		tmp = R * (2.0 * math.atan2(t_3, math.sqrt((1.0 - (t_0 + (math.cos(phi2) * (math.cos(phi1) * math.pow(math.sin((lambda1 * 0.5)), 2.0))))))))
	else:
		tmp = R * (2.0 * math.atan2(t_3, math.sqrt((1.0 - (t_0 + (t_2 * math.pow(math.sin((lambda2 * -0.5)), 2.0)))))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(cos(phi1) * cos(phi2))
	t_3 = sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_1 * Float64(t_2 * t_1))))
	tmp = 0.0
	if (lambda2 <= 1e-5)
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * (sin(Float64(lambda1 * 0.5)) ^ 2.0)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - Float64(t_0 + Float64(t_2 * (sin(Float64(lambda2 * -0.5)) ^ 2.0))))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) * 0.5)) ^ 2.0;
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = cos(phi1) * cos(phi2);
	t_3 = sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_1 * (t_2 * t_1))));
	tmp = 0.0;
	if (lambda2 <= 1e-5)
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - (t_0 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5)) ^ 2.0))))))));
	else
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - (t_0 + (t_2 * (sin((lambda2 * -0.5)) ^ 2.0)))))));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.00000000000000008e-5

   1. Initial program 64.2%

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

   2. Taylor expanded in lambda2 around 0 56.5%

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

   if 1.00000000000000008e-5 < lambda2

   1. Initial program 49.1%

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

   2. Taylor expanded in lambda1 around 0 49.0%

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

4. Final simplification 54.9%

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

Alternative 6: 56.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{-8} \lor \neg \left(\phi_1 \leq 2.6 \cdot 10^{-23}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot t_0}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t_0, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_0}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)))
   (if (or (<= phi1 -8e-8) (not (<= phi1 2.6e-23)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (* phi1 0.5)) 2.0) (* (cos phi1) t_0)))
        (sqrt
         (-
          1.0
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (*
            (* (* (cos phi1) (cos phi2)) (sin (/ (- lambda1 lambda2) 2.0)))
            (sin (* lambda2 -0.5)))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma (cos phi2) t_0 (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double tmp;
	if ((phi1 <= -8e-8) || !(phi1 <= 2.6e-23)) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * t_0))), sqrt((1.0 - (pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (((cos(phi1) * cos(phi2)) * sin(((lambda1 - lambda2) / 2.0))) * sin((lambda2 * -0.5))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), t_0, pow(sin((phi2 * -0.5)), 2.0))), sqrt((pow(cos((phi2 * -0.5)), 2.0) - (cos(phi2) * t_0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	tmp = 0.0
	if ((phi1 <= -8e-8) || !(phi1 <= 2.6e-23))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * t_0))), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(Float64(Float64(cos(phi1) * cos(phi2)) * sin(Float64(Float64(lambda1 - lambda2) / 2.0))) * sin(Float64(lambda2 * -0.5)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), t_0, (sin(Float64(phi2 * -0.5)) ^ 2.0))), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - Float64(cos(phi2) * t_0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi1, -8e-8], N[Not[LessEqual[phi1, 2.6e-23]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + 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]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -8.0000000000000002e-8 or 2.6e-23 < phi1

   1. Initial program 47.2%

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

   2. Taylor expanded in lambda1 around 0 37.9%

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

   3. Taylor expanded in phi2 around 0 39.0%

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

   if -8.0000000000000002e-8 < phi1 < 2.6e-23

   1. Initial program 79.8%

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

   2. Taylor expanded in phi1 around 0 79.8%

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

      1. associate--r+ 79.8%

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

      2. unpow2 79.8%

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

      3. 1-sub-sin 79.9%

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

      4. unpow2 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in phi1 around 0 79.1%

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

      1. +-commutative 79.1%

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

      2. fma-def 79.1%

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

   7. Simplified 79.1%

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

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{-8} \lor \neg \left(\phi_1 \leq 2.6 \cdot 10^{-23}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 7: 57.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_3 := \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\\ \mathbf{if}\;\phi_1 \leq -2200 \lor \neg \left(\phi_1 \leq 1.55 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot t_2}}{\sqrt{1 - \left(t_0 + t_3 \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot t_3}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_3 (* (* (cos phi1) (cos phi2)) t_1)))
   (if (or (<= phi1 -2200.0) (not (<= phi1 1.55e-5)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (* phi1 0.5)) 2.0) (* (cos phi1) t_2)))
        (sqrt (- 1.0 (+ t_0 (* t_3 (sin (* lambda2 -0.5)))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* t_1 t_3)))
        (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_3 = (cos(phi1) * cos(phi2)) * t_1;
	double tmp;
	if ((phi1 <= -2200.0) || !(phi1 <= 1.55e-5)) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * t_2))), sqrt((1.0 - (t_0 + (t_3 * sin((lambda2 * -0.5))))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_1 * t_3))), sqrt((pow(cos((phi2 * -0.5)), 2.0) - (cos(phi2) * 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 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    t_3 = (cos(phi1) * cos(phi2)) * t_1
    if ((phi1 <= (-2200.0d0)) .or. (.not. (phi1 <= 1.55d-5))) then
        tmp = r * (2.0d0 * atan2(sqrt(((sin((phi1 * 0.5d0)) ** 2.0d0) + (cos(phi1) * t_2))), sqrt((1.0d0 - (t_0 + (t_3 * sin((lambda2 * (-0.5d0)))))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (t_1 * t_3))), sqrt(((cos((phi2 * (-0.5d0))) ** 2.0d0) - (cos(phi2) * 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.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_3 = (Math.cos(phi1) * Math.cos(phi2)) * t_1;
	double tmp;
	if ((phi1 <= -2200.0) || !(phi1 <= 1.55e-5)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((phi1 * 0.5)), 2.0) + (Math.cos(phi1) * t_2))), Math.sqrt((1.0 - (t_0 + (t_3 * Math.sin((lambda2 * -0.5))))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (t_1 * t_3))), Math.sqrt((Math.pow(Math.cos((phi2 * -0.5)), 2.0) - (Math.cos(phi2) * t_2)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	t_3 = (math.cos(phi1) * math.cos(phi2)) * t_1
	tmp = 0
	if (phi1 <= -2200.0) or not (phi1 <= 1.55e-5):
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((phi1 * 0.5)), 2.0) + (math.cos(phi1) * t_2))), math.sqrt((1.0 - (t_0 + (t_3 * math.sin((lambda2 * -0.5))))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (t_1 * t_3))), math.sqrt((math.pow(math.cos((phi2 * -0.5)), 2.0) - (math.cos(phi2) * t_2)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_3 = Float64(Float64(cos(phi1) * cos(phi2)) * t_1)
	tmp = 0.0
	if ((phi1 <= -2200.0) || !(phi1 <= 1.55e-5))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * t_2))), sqrt(Float64(1.0 - Float64(t_0 + Float64(t_3 * sin(Float64(lambda2 * -0.5)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(t_1 * t_3))), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - Float64(cos(phi2) * t_2))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	t_3 = (cos(phi1) * cos(phi2)) * t_1;
	tmp = 0.0;
	if ((phi1 <= -2200.0) || ~((phi1 <= 1.55e-5)))
		tmp = R * (2.0 * atan2(sqrt(((sin((phi1 * 0.5)) ^ 2.0) + (cos(phi1) * t_2))), sqrt((1.0 - (t_0 + (t_3 * sin((lambda2 * -0.5))))))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_1 * t_3))), sqrt(((cos((phi2 * -0.5)) ^ 2.0) - (cos(phi2) * t_2)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, If[Or[LessEqual[phi1, -2200.0], N[Not[LessEqual[phi1, 1.55e-5]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$0 + N[(t$95$3 * N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$1 * t$95$3), $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$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2200 or 1.55000000000000007e-5 < phi1

   1. Initial program 45.8%

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

   2. Taylor expanded in lambda1 around 0 36.9%

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

   3. Taylor expanded in phi2 around 0 38.2%

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

   if -2200 < phi1 < 1.55000000000000007e-5

   1. Initial program 79.6%

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

   2. Taylor expanded in phi1 around 0 79.0%

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

      1. associate--r+ 79.1%

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

      2. unpow2 79.1%

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

      3. 1-sub-sin 79.1%

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

      4. unpow2 79.1%

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

   4. Simplified 79.1%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2200 \lor \neg \left(\phi_1 \leq 1.55 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{1 - \left({\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) \cdot \sin \left(\lambda_2 \cdot -0.5\right)\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 8: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, t_0, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= (- lambda1 lambda2) -0.002)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (fma (cos phi2) t_0 (pow (sin (* phi2 -0.5)) 2.0)))
        (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) (* (cos phi2) t_0))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* t_1 (* (* (cos phi1) (cos phi2)) t_1))))
        (sqrt (- 1.0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((lambda1 - lambda2) <= -0.002) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi2), t_0, 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((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * ((cos(phi1) * cos(phi2)) * t_1)))), sqrt((1.0 - pow(sin(((phi1 - phi2) * 0.5)), 2.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -0.002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi2), t_0, (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((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)))), sqrt(Float64(1.0 - (sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -0.002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi2], $MachinePrecision] * t$95$0 + 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[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2e-3

   1. Initial program 58.1%

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

   2. Taylor expanded in phi1 around 0 43.7%

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

      1. associate--r+ 43.7%

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

      2. unpow2 43.7%

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

      3. 1-sub-sin 43.8%

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

      4. unpow2 43.8%

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

   4. Simplified 43.8%

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

   5. Taylor expanded in phi1 around 0 43.4%

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

      1. +-commutative 43.4%

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

      2. fma-def 43.4%

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

   7. Simplified 43.4%

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

   if -2e-3 < (-.f64 lambda1 lambda2)

   1. Initial program 62.4%

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

   2. Taylor expanded in lambda1 around 0 52.5%

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

   3. Taylor expanded in lambda2 around 0 43.4%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_2, {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}, {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 9: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + t_0}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_1\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi2) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (<= (- lambda1 lambda2) -0.002)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (* phi2 -0.5)) 2.0) t_0))
        (sqrt (- (pow (cos (* phi2 -0.5)) 2.0) t_0)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
          (* t_1 (* (* (cos phi1) (cos phi2)) t_1))))
        (sqrt (- 1.0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos(phi2) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((lambda1 - lambda2) <= -0.002) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((phi2 * -0.5)), 2.0) + t_0)), sqrt((pow(cos((phi2 * -0.5)), 2.0) - t_0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * ((cos(phi1) * cos(phi2)) * t_1)))), sqrt((1.0 - pow(sin(((phi1 - phi2) * 0.5)), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    if ((lambda1 - lambda2) <= (-0.002d0)) then
        tmp = r * (2.0d0 * atan2(sqrt(((sin((phi2 * (-0.5d0))) ** 2.0d0) + t_0)), sqrt(((cos((phi2 * (-0.5d0))) ** 2.0d0) - t_0))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_1 * ((cos(phi1) * cos(phi2)) * t_1)))), sqrt((1.0d0 - (sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos(phi2) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((lambda1 - lambda2) <= -0.002) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((phi2 * -0.5)), 2.0) + t_0)), Math.sqrt((Math.pow(Math.cos((phi2 * -0.5)), 2.0) - t_0))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * ((Math.cos(phi1) * Math.cos(phi2)) * t_1)))), Math.sqrt((1.0 - Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos(phi2) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if (lambda1 - lambda2) <= -0.002:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((phi2 * -0.5)), 2.0) + t_0)), math.sqrt((math.pow(math.cos((phi2 * -0.5)), 2.0) - t_0))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * ((math.cos(phi1) * math.cos(phi2)) * t_1)))), math.sqrt((1.0 - math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi2) * (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -0.002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi2 * -0.5)) ^ 2.0) + t_0)), sqrt(Float64((cos(Float64(phi2 * -0.5)) ^ 2.0) - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_1 * Float64(Float64(cos(phi1) * cos(phi2)) * t_1)))), sqrt(Float64(1.0 - (sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(phi2) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0);
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -0.002)
		tmp = R * (2.0 * atan2(sqrt(((sin((phi2 * -0.5)) ^ 2.0) + t_0)), sqrt(((cos((phi2 * -0.5)) ^ 2.0) - t_0))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_1 * ((cos(phi1) * cos(phi2)) * t_1)))), sqrt((1.0 - (sin(((phi1 - phi2) * 0.5)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -0.002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(N[Power[N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2e-3

   1. Initial program 58.1%

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

   2. Taylor expanded in phi1 around 0 43.7%

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

      1. associate--r+ 43.7%

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

      2. unpow2 43.7%

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

      3. 1-sub-sin 43.8%

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

      4. unpow2 43.8%

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

   4. Simplified 43.8%

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

   5. Taylor expanded in phi1 around 0 43.4%

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

   if -2e-3 < (-.f64 lambda1 lambda2)

   1. Initial program 62.4%

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

   2. Taylor expanded in lambda1 around 0 52.5%

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

   3. Taylor expanded in lambda2 around 0 43.4%

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

4. Final simplification 43.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2} - \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 10: 37.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot t_0\right)\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* t_0 (* (* (cos phi1) (cos phi2)) t_0))))
   (if (<= (- lambda1 lambda2) -0.002)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_1 (pow (sin (* phi1 0.5)) 2.0)))
        (sqrt (- 1.0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_1))
        (sqrt (- 1.0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = t_0 * ((cos(phi1) * cos(phi2)) * t_0);
	double tmp;
	if ((lambda1 - lambda2) <= -0.002) {
		tmp = R * (2.0 * atan2(sqrt((t_1 + pow(sin((phi1 * 0.5)), 2.0))), sqrt((1.0 - pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), sqrt((1.0 - pow(sin(((phi1 - phi2) * 0.5)), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = t_0 * ((cos(phi1) * cos(phi2)) * t_0)
    if ((lambda1 - lambda2) <= (-0.002d0)) then
        tmp = r * (2.0d0 * atan2(sqrt((t_1 + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt((1.0d0 - (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + t_1)), sqrt((1.0d0 - (sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = t_0 * ((Math.cos(phi1) * Math.cos(phi2)) * t_0);
	double tmp;
	if ((lambda1 - lambda2) <= -0.002) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_1 + Math.pow(Math.sin((phi1 * 0.5)), 2.0))), Math.sqrt((1.0 - Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	} else {
		tmp = 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(((phi1 - phi2) * 0.5)), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = t_0 * ((math.cos(phi1) * math.cos(phi2)) * t_0)
	tmp = 0
	if (lambda1 - lambda2) <= -0.002:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_1 + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((1.0 - math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))))
	else:
		tmp = 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(((phi1 - phi2) * 0.5)), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(t_0 * Float64(Float64(cos(phi1) * cos(phi2)) * t_0))
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -0.002)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + (sin(Float64(phi1 * 0.5)) ^ 2.0))), sqrt(Float64(1.0 - (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), sqrt(Float64(1.0 - (sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = t_0 * ((cos(phi1) * cos(phi2)) * t_0);
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -0.002)
		tmp = R * (2.0 * atan2(sqrt((t_1 + (sin((phi1 * 0.5)) ^ 2.0))), sqrt((1.0 - (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), sqrt((1.0 - (sin(((phi1 - phi2) * 0.5)) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -0.002], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -2e-3

   1. Initial program 58.1%

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

   2. Taylor expanded in phi1 around 0 43.7%

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

      1. associate--r+ 43.7%

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

      2. unpow2 43.7%

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

      3. 1-sub-sin 43.8%

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

      4. unpow2 43.8%

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

   4. Simplified 43.8%

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

   5. Taylor expanded in phi2 around 0 30.7%

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

   6. Taylor expanded in phi2 around 0 30.5%

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

   if -2e-3 < (-.f64 lambda1 lambda2)

   1. Initial program 62.4%

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

   2. Taylor expanded in lambda1 around 0 52.5%

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

   3. Taylor expanded in lambda2 around 0 43.4%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -0.002:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 11: 35.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \cos \phi_1 \cdot \cos \phi_2\\ t_2 := {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.24 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_1 \cdot t_0\right) + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 + t_1 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - t_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* (cos phi1) (cos phi2)))
        (t_2 (pow (sin (* (- phi1 phi2) 0.5)) 2.0)))
   (if (<= (- lambda1 lambda2) -1.24e-45)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* t_0 (* t_1 t_0)) (pow (sin (* phi1 0.5)) 2.0)))
        (sqrt (- 1.0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_2 (* t_1 (pow (sin (* lambda2 -0.5)) 2.0))))
        (sqrt (- 1.0 t_2))))))))

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);
	double t_2 = pow(sin(((phi1 - phi2) * 0.5)), 2.0);
	double tmp;
	if ((lambda1 - lambda2) <= -1.24e-45) {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_1 * t_0)) + pow(sin((phi1 * 0.5)), 2.0))), sqrt((1.0 - pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_2 + (t_1 * pow(sin((lambda2 * -0.5)), 2.0)))), sqrt((1.0 - 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 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = cos(phi1) * cos(phi2)
    t_2 = sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0
    if ((lambda1 - lambda2) <= (-1.24d-45)) then
        tmp = r * (2.0d0 * atan2(sqrt(((t_0 * (t_1 * t_0)) + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt((1.0d0 - (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_2 + (t_1 * (sin((lambda2 * (-0.5d0))) ** 2.0d0)))), sqrt((1.0d0 - 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.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.cos(phi1) * Math.cos(phi2);
	double t_2 = Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0);
	double tmp;
	if ((lambda1 - lambda2) <= -1.24e-45) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_0 * (t_1 * t_0)) + Math.pow(Math.sin((phi1 * 0.5)), 2.0))), Math.sqrt((1.0 - Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_2 + (t_1 * Math.pow(Math.sin((lambda2 * -0.5)), 2.0)))), Math.sqrt((1.0 - t_2))));
	}
	return tmp;
}

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_2 = math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)
	tmp = 0
	if (lambda1 - lambda2) <= -1.24e-45:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_0 * (t_1 * t_0)) + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((1.0 - math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_2 + (t_1 * math.pow(math.sin((lambda2 * -0.5)), 2.0)))), math.sqrt((1.0 - t_2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(cos(phi1) * cos(phi2))
	t_2 = sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1.24e-45)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(t_1 * t_0)) + (sin(Float64(phi1 * 0.5)) ^ 2.0))), sqrt(Float64(1.0 - (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_2 + Float64(t_1 * (sin(Float64(lambda2 * -0.5)) ^ 2.0)))), sqrt(Float64(1.0 - t_2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = cos(phi1) * cos(phi2);
	t_2 = sin(((phi1 - phi2) * 0.5)) ^ 2.0;
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1.24e-45)
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_1 * t_0)) + (sin((phi1 * 0.5)) ^ 2.0))), sqrt((1.0 - (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_2 + (t_1 * (sin((lambda2 * -0.5)) ^ 2.0)))), sqrt((1.0 - t_2))));
	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[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1.24e-45], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(t$95$1 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$2 + N[(t$95$1 * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -1.24e-45

   1. Initial program 59.4%

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

   2. Taylor expanded in phi1 around 0 45.7%

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

      1. associate--r+ 45.8%

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

      2. unpow2 45.8%

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

      3. 1-sub-sin 45.8%

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

      4. unpow2 45.8%

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

   4. Simplified 45.8%

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

   5. Taylor expanded in phi2 around 0 33.0%

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

   6. Taylor expanded in phi2 around 0 32.9%

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

   if -1.24e-45 < (-.f64 lambda1 lambda2)

   1. Initial program 61.7%

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

   2. Taylor expanded in lambda1 around 0 51.5%

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

   3. Taylor expanded in lambda2 around 0 42.1%

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

   4. Taylor expanded in lambda1 around 0 39.5%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1.24 \cdot 10^{-45}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\sin \left(\frac{\lambda_1 - \lambda_2}{2}\right) \cdot \left(\left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\right) + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 12: 32.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sqrt{1 - t_0}\\ \mathbf{if}\;\lambda_2 \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)}}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0)) (t_1 (sqrt (- 1.0 t_0))))
   (if (<= lambda2 1.25e-52)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_0 (* (cos phi2) (* (cos phi1) (pow (sin (* lambda1 0.5)) 2.0)))))
        t_1)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (pow (sin (* phi1 0.5)) 2.0)
          (* (cos phi1) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))
        t_1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_1 = sqrt((1.0 - t_0));
	double tmp;
	if (lambda2 <= 1.25e-52) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi2) * (cos(phi1) * pow(sin((lambda1 * 0.5)), 2.0))))), t_1));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))), 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(((phi1 - phi2) * 0.5d0)) ** 2.0d0
    t_1 = sqrt((1.0d0 - t_0))
    if (lambda2 <= 1.25d-52) then
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5d0)) ** 2.0d0))))), t_1))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin((phi1 * 0.5d0)) ** 2.0d0) + (cos(phi1) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)))), 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.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_1 = Math.sqrt((1.0 - t_0));
	double tmp;
	if (lambda2 <= 1.25e-52) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.pow(Math.sin((lambda1 * 0.5)), 2.0))))), t_1));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((phi1 * 0.5)), 2.0) + (Math.cos(phi1) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))), t_1));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)
	t_1 = math.sqrt((1.0 - t_0))
	tmp = 0
	if lambda2 <= 1.25e-52:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.pow(math.sin((lambda1 * 0.5)), 2.0))))), t_1))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((phi1 * 0.5)), 2.0) + (math.cos(phi1) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))), t_1))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0
	t_1 = sqrt(Float64(1.0 - t_0))
	tmp = 0.0
	if (lambda2 <= 1.25e-52)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * (sin(Float64(lambda1 * 0.5)) ^ 2.0))))), t_1)));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0)))), t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) * 0.5)) ^ 2.0;
	t_1 = sqrt((1.0 - t_0));
	tmp = 0.0;
	if (lambda2 <= 1.25e-52)
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5)) ^ 2.0))))), t_1));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin((phi1 * 0.5)) ^ 2.0) + (cos(phi1) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)))), t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, 1.25e-52], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.25e-52

   1. Initial program 63.8%

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

   2. Taylor expanded in lambda1 around 0 46.5%

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

   3. Taylor expanded in lambda2 around 0 38.7%

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

   4. Taylor expanded in lambda2 around 0 36.9%

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

   if 1.25e-52 < lambda2

   1. Initial program 51.8%

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

   2. Taylor expanded in lambda1 around 0 50.0%

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

   3. Taylor expanded in lambda2 around 0 25.8%

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

   4. Taylor expanded in phi2 around 0 26.3%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.25 \cdot 10^{-52}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 13: 32.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sqrt{1 - t_0}\\ \mathbf{if}\;\lambda_2 \leq 5.9 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)}}{t_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0)) (t_1 (sqrt (- 1.0 t_0))))
   (if (<= lambda2 5.9e-80)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+ t_0 (* (cos phi2) (* (cos phi1) (pow (sin (* lambda1 0.5)) 2.0)))))
        t_1)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          t_0
          (* (* (cos phi1) (cos phi2)) (pow (sin (* lambda2 -0.5)) 2.0))))
        t_1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_1 = sqrt((1.0 - t_0));
	double tmp;
	if (lambda2 <= 5.9e-80) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi2) * (cos(phi1) * pow(sin((lambda1 * 0.5)), 2.0))))), t_1));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + ((cos(phi1) * cos(phi2)) * pow(sin((lambda2 * -0.5)), 2.0)))), 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(((phi1 - phi2) * 0.5d0)) ** 2.0d0
    t_1 = sqrt((1.0d0 - t_0))
    if (lambda2 <= 5.9d-80) then
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5d0)) ** 2.0d0))))), t_1))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + ((cos(phi1) * cos(phi2)) * (sin((lambda2 * (-0.5d0))) ** 2.0d0)))), 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.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0);
	double t_1 = Math.sqrt((1.0 - t_0));
	double tmp;
	if (lambda2 <= 5.9e-80) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (Math.cos(phi2) * (Math.cos(phi1) * Math.pow(Math.sin((lambda1 * 0.5)), 2.0))))), t_1));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + ((Math.cos(phi1) * Math.cos(phi2)) * Math.pow(Math.sin((lambda2 * -0.5)), 2.0)))), t_1));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)
	t_1 = math.sqrt((1.0 - t_0))
	tmp = 0
	if lambda2 <= 5.9e-80:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (math.cos(phi2) * (math.cos(phi1) * math.pow(math.sin((lambda1 * 0.5)), 2.0))))), t_1))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + ((math.cos(phi1) * math.cos(phi2)) * math.pow(math.sin((lambda2 * -0.5)), 2.0)))), t_1))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0
	t_1 = sqrt(Float64(1.0 - t_0))
	tmp = 0.0
	if (lambda2 <= 5.9e-80)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(cos(phi2) * Float64(cos(phi1) * (sin(Float64(lambda1 * 0.5)) ^ 2.0))))), t_1)));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(Float64(cos(phi1) * cos(phi2)) * (sin(Float64(lambda2 * -0.5)) ^ 2.0)))), t_1)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) * 0.5)) ^ 2.0;
	t_1 = sqrt((1.0 - t_0));
	tmp = 0.0;
	if (lambda2 <= 5.9e-80)
		tmp = R * (2.0 * atan2(sqrt((t_0 + (cos(phi2) * (cos(phi1) * (sin((lambda1 * 0.5)) ^ 2.0))))), t_1));
	else
		tmp = R * (2.0 * atan2(sqrt((t_0 + ((cos(phi1) * cos(phi2)) * (sin((lambda2 * -0.5)) ^ 2.0)))), t_1));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, 5.9e-80], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision] * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 5.9000000000000001e-80

   1. Initial program 64.0%

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

   2. Taylor expanded in lambda1 around 0 46.3%

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

   3. Taylor expanded in lambda2 around 0 38.4%

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

   4. Taylor expanded in lambda2 around 0 36.5%

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

   if 5.9000000000000001e-80 < lambda2

   1. Initial program 52.0%

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

   2. Taylor expanded in lambda1 around 0 50.3%

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

   3. Taylor expanded in lambda2 around 0 27.5%

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

   4. Taylor expanded in lambda1 around 0 28.1%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 5.9 \cdot 10^{-80}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \cos \phi_2 \cdot \left(\cos \phi_1 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2} + \left(\cos \phi_1 \cdot \cos \phi_2\right) \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 14: 33.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -2.65 \cdot 10^{-8} \lor \neg \left(\phi_1 \leq 1.2 \cdot 10^{-65}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot t_0}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot t_0}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)))
   (if (or (<= phi1 -2.65e-8) (not (<= phi1 1.2e-65)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (* phi1 0.5)) 2.0) (* (cos phi1) t_0)))
        (sqrt (- 1.0 (pow (sin (* (- phi1 phi2) 0.5)) 2.0))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (* phi2 -0.5)) 2.0) (* (cos phi2) t_0)))
        (sqrt (pow (cos (* phi2 -0.5)) 2.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double tmp;
	if ((phi1 <= -2.65e-8) || !(phi1 <= 1.2e-65)) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * t_0))), sqrt((1.0 - pow(sin(((phi1 - phi2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((phi2 * -0.5)), 2.0) + (cos(phi2) * t_0))), sqrt(pow(cos((phi2 * -0.5)), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    if ((phi1 <= (-2.65d-8)) .or. (.not. (phi1 <= 1.2d-65))) then
        tmp = r * (2.0d0 * atan2(sqrt(((sin((phi1 * 0.5d0)) ** 2.0d0) + (cos(phi1) * t_0))), sqrt((1.0d0 - (sin(((phi1 - phi2) * 0.5d0)) ** 2.0d0)))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin((phi2 * (-0.5d0))) ** 2.0d0) + (cos(phi2) * t_0))), sqrt((cos((phi2 * (-0.5d0))) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double tmp;
	if ((phi1 <= -2.65e-8) || !(phi1 <= 1.2e-65)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((phi1 * 0.5)), 2.0) + (Math.cos(phi1) * t_0))), Math.sqrt((1.0 - Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((phi2 * -0.5)), 2.0) + (Math.cos(phi2) * t_0))), Math.sqrt(Math.pow(Math.cos((phi2 * -0.5)), 2.0))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	tmp = 0
	if (phi1 <= -2.65e-8) or not (phi1 <= 1.2e-65):
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((phi1 * 0.5)), 2.0) + (math.cos(phi1) * t_0))), math.sqrt((1.0 - math.pow(math.sin(((phi1 - phi2) * 0.5)), 2.0)))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((phi2 * -0.5)), 2.0) + (math.cos(phi2) * t_0))), math.sqrt(math.pow(math.cos((phi2 * -0.5)), 2.0))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	tmp = 0.0
	if ((phi1 <= -2.65e-8) || !(phi1 <= 1.2e-65))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * t_0))), sqrt(Float64(1.0 - (sin(Float64(Float64(phi1 - phi2) * 0.5)) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(phi2 * -0.5)) ^ 2.0) + Float64(cos(phi2) * t_0))), sqrt((cos(Float64(phi2 * -0.5)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	tmp = 0.0;
	if ((phi1 <= -2.65e-8) || ~((phi1 <= 1.2e-65)))
		tmp = R * (2.0 * atan2(sqrt(((sin((phi1 * 0.5)) ^ 2.0) + (cos(phi1) * t_0))), sqrt((1.0 - (sin(((phi1 - phi2) * 0.5)) ^ 2.0)))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin((phi2 * -0.5)) ^ 2.0) + (cos(phi2) * t_0))), sqrt((cos((phi2 * -0.5)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[phi1, -2.65e-8], N[Not[LessEqual[phi1, 1.2e-65]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Power[N[Cos[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.6499999999999999e-8 or 1.2000000000000001e-65 < phi1

   1. Initial program 48.7%

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

   2. Taylor expanded in lambda1 around 0 38.4%

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

   3. Taylor expanded in lambda2 around 0 29.9%

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

   4. Taylor expanded in phi2 around 0 31.0%

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

   if -2.6499999999999999e-8 < phi1 < 1.2000000000000001e-65

   1. Initial program 79.5%

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

   2. Taylor expanded in lambda1 around 0 61.2%

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

   3. Taylor expanded in lambda2 around 0 44.2%

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

   4. Taylor expanded in phi1 around 0 44.2%

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

   5. Taylor expanded in phi1 around 0 44.2%

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

      1. *-commutative 44.2%

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

      2. unpow2 44.2%

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

      3. 1-sub-sin 44.2%

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

      4. unpow2 44.2%

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

      5. *-commutative 44.2%

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

   7. Simplified 44.2%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.65 \cdot 10^{-8} \lor \neg \left(\phi_1 \leq 1.2 \cdot 10^{-65}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{1 - {\sin \left(\left(\phi_1 - \phi_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\phi_2 \cdot -0.5\right)}^{2} + \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}{\sqrt{{\cos \left(\phi_2 \cdot -0.5\right)}^{2}}}\right)\\ \end{array} \]

Alternative 15: 27.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (2.0d0 * atan2(sqrt(((sin((phi2 * (-0.5d0))) ** 2.0d0) + (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)))), sqrt((1.0d0 - (sin(((phi1 - phi2) * 0.5d0)) ** 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(Math.sqrt((Math.pow(Math.sin((phi2 * -0.5)), 2.0) + (Math.cos(phi2) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))), Math.sqrt((1.0 - Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

2. Taylor expanded in lambda1 around 0 47.4%

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

3. Taylor expanded in lambda2 around 0 35.5%

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

4. Taylor expanded in phi1 around 0 28.9%

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

5. Final simplification 28.9%

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

Alternative 16: 27.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

2. Taylor expanded in lambda1 around 0 47.4%

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

3. Taylor expanded in lambda2 around 0 35.5%

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

4. Taylor expanded in phi1 around 0 28.9%

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

5. Taylor expanded in phi1 around 0 28.9%

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

   1. *-commutative 28.9%

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

   2. unpow2 28.9%

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

   3. 1-sub-sin 28.9%

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

   4. unpow2 28.9%

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

   5. *-commutative 28.9%

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

7. Simplified 28.9%

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

8. Final simplification 28.9%

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

Alternative 17: 11.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (2.0d0 * atan2(sin(((lambda1 - lambda2) * 0.5d0)), sqrt((1.0d0 - (sin(((phi1 - phi2) * 0.5d0)) ** 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(Math.sin(((lambda1 - lambda2) * 0.5)), Math.sqrt((1.0 - Math.pow(Math.sin(((phi1 - phi2) * 0.5)), 2.0)))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.9%

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

2. Taylor expanded in lambda1 around 0 47.4%

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

3. Taylor expanded in lambda2 around 0 35.5%

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

4. Taylor expanded in phi1 around 0 28.9%

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

5. Taylor expanded in phi2 around 0 12.7%

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

6. Final simplification 12.7%

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

Reproduce

herbie shell --seed 2023230 
(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))))))))))