Distance on a great circle

Percentage Accurate: 61.8% → 78.6%

Time: 2.4min

Alternatives: 23

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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 2: 78.0% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 3: 78.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* t_0 t_0))
        (t_2 (* (sin (/ phi1 2.0)) (cos (/ phi2 2.0))))
        (t_3 (* (cos (/ phi1 2.0)) (sin (/ phi2 2.0)))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt (fma (cos phi1) (* (cos phi2) t_1) (pow (- t_2 t_3) 2.0)))
      (sqrt
       (- 1.0 (fma (cos phi2) (* (cos phi1) t_1) (pow (- t_3 t_2) 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 * t_0;
	double t_2 = sin((phi1 / 2.0)) * cos((phi2 / 2.0));
	double t_3 = cos((phi1 / 2.0)) * sin((phi2 / 2.0));
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * t_1), pow((t_2 - t_3), 2.0))), sqrt((1.0 - fma(cos(phi2), (cos(phi1) * t_1), pow((t_3 - t_2), 2.0))))));
}

Julia

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

Wolfram

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

Derivation

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1. Add Preprocessing

Alternative 4: 78.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 5: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ t_2 := t_0 \cdot t_0\\ t_3 := \cos \left(\frac{\phi_1}{2}\right) \cdot \sin \left(\frac{\phi_2}{2}\right)\\ t_4 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_5 := \sin \left(\frac{\phi_1}{2}\right) \cdot \cos \left(\frac{\phi_2}{2}\right)\\ t_6 := \sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot t_2, {\left(t_3 - t_5\right)}^{2}\right)}\\ t_7 := {\left(t_5 - t_3\right)}^{2}\\ \mathbf{if}\;\lambda_2 \leq -1.75 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot t_2, t_4\right)}}{t_6}\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.0018:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_7 + t_1 \cdot \sin \left(\lambda_1 \cdot 0.5\right)}}{\sqrt{1 - \left(t_7 + t_0 \cdot t_1\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}, t_4\right)}}{t_6}\right)\\ \end{array} \end{array} \]

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_2 (* t_0 t_0))
        (t_3 (* (cos (/ phi1 2.0)) (sin (/ phi2 2.0))))
        (t_4 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_5 (* (sin (/ phi1 2.0)) (cos (/ phi2 2.0))))
        (t_6
         (sqrt
          (- 1.0 (fma (cos phi2) (* (cos phi1) t_2) (pow (- t_3 t_5) 2.0)))))
        (t_7 (pow (- t_5 t_3) 2.0)))
   (if (<= lambda2 -1.75e-7)
     (* R (* 2.0 (atan2 (sqrt (fma (cos phi1) (* (cos phi2) t_2) t_4)) t_6)))
     (if (<= lambda2 0.0018)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ t_7 (* t_1 (sin (* lambda1 0.5)))))
          (sqrt (- 1.0 (+ t_7 (* t_0 t_1)))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (fma
            (cos phi1)
            (* (cos phi2) (pow (sin (* lambda2 -0.5)) 2.0))
            t_4))
          t_6)))))))

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));
	double t_2 = t_0 * t_0;
	double t_3 = cos((phi1 / 2.0)) * sin((phi2 / 2.0));
	double t_4 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_5 = sin((phi1 / 2.0)) * cos((phi2 / 2.0));
	double t_6 = sqrt((1.0 - fma(cos(phi2), (cos(phi1) * t_2), pow((t_3 - t_5), 2.0))));
	double t_7 = pow((t_5 - t_3), 2.0);
	double tmp;
	if (lambda2 <= -1.75e-7) {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * t_2), t_4)), t_6));
	} else if (lambda2 <= 0.0018) {
		tmp = R * (2.0 * atan2(sqrt((t_7 + (t_1 * sin((lambda1 * 0.5))))), sqrt((1.0 - (t_7 + (t_0 * t_1))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * pow(sin((lambda2 * -0.5)), 2.0)), t_4)), t_6));
	}
	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(cos(phi1) * cos(phi2)))
	t_2 = Float64(t_0 * t_0)
	t_3 = Float64(cos(Float64(phi1 / 2.0)) * sin(Float64(phi2 / 2.0)))
	t_4 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_5 = Float64(sin(Float64(phi1 / 2.0)) * cos(Float64(phi2 / 2.0)))
	t_6 = sqrt(Float64(1.0 - fma(cos(phi2), Float64(cos(phi1) * t_2), (Float64(t_3 - t_5) ^ 2.0))))
	t_7 = Float64(t_5 - t_3) ^ 2.0
	tmp = 0.0
	if (lambda2 <= -1.75e-7)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * t_2), t_4)), t_6)));
	elseif (lambda2 <= 0.0018)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_7 + Float64(t_1 * sin(Float64(lambda1 * 0.5))))), sqrt(Float64(1.0 - Float64(t_7 + Float64(t_0 * t_1)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * (sin(Float64(lambda2 * -0.5)) ^ 2.0)), t_4)), t_6)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$0), $MachinePrecision]}, Block[{t$95$3 = N[(N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[(N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * t$95$2), $MachinePrecision] + N[Power[N[(t$95$3 - t$95$5), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$7 = N[Power[N[(t$95$5 - t$95$3), $MachinePrecision], 2.0], $MachinePrecision]}, If[LessEqual[lambda2, -1.75e-7], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * t$95$2), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.0018], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$7 + N[(t$95$1 * N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(t$95$7 + N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$6], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 6: 78.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (+
          (pow
           (-
            (* (sin (/ phi1 2.0)) (cos (/ phi2 2.0)))
            (* (cos (/ phi1 2.0)) (sin (/ phi2 2.0))))
           2.0)
          (* t_0 (* t_0 (* (cos phi1) (cos phi2)))))))
   (* 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 / 2.0)) * cos((phi2 / 2.0))) - (cos((phi1 / 2.0)) * sin((phi2 / 2.0)))), 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))));
	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 / 2.0d0)) * cos((phi2 / 2.0d0))) - (cos((phi1 / 2.0d0)) * sin((phi2 / 2.0d0)))) ** 2.0d0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))
    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 / 2.0)) * Math.cos((phi2 / 2.0))) - (Math.cos((phi1 / 2.0)) * Math.sin((phi2 / 2.0)))), 2.0) + (t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2))));
	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 / 2.0)) * math.cos((phi2 / 2.0))) - (math.cos((phi1 / 2.0)) * math.sin((phi2 / 2.0)))), 2.0) + (t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2))))
	return R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))

Julia

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 7: 62.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 8: 62.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 9: 62.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 10: 43.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ \mathbf{if}\;t_0 \leq -0.018 \lor \neg \left(t_0 \leq 0.07\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_1 \cdot t_1 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_2 \cdot t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)))
   (if (or (<= t_0 -0.018) (not (<= t_0 0.07)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* (cos phi1) t_1) (pow (sin (* phi1 0.5)) 2.0)))
        (sqrt (- 1.0 (+ (* (cos phi2) t_1) (pow (sin (* phi2 -0.5)) 2.0)))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_0 (* t_0 (* (cos phi1) (cos phi2))))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double tmp;
	if ((t_0 <= -0.018) || !(t_0 <= 0.07)) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * t_1) + pow(sin((phi1 * 0.5)), 2.0))), sqrt((1.0 - ((cos(phi2) * t_1) + pow(sin((phi2 * -0.5)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    if ((t_0 <= (-0.018d0)) .or. (.not. (t_0 <= 0.07d0))) then
        tmp = r * (2.0d0 * atan2(sqrt(((cos(phi1) * t_1) + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt((1.0d0 - ((cos(phi2) * t_1) + (sin((phi2 * (-0.5d0))) ** 2.0d0))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double tmp;
	if ((t_0 <= -0.018) || !(t_0 <= 0.07)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((Math.cos(phi1) * t_1) + Math.pow(Math.sin((phi1 * 0.5)), 2.0))), Math.sqrt((1.0 - ((Math.cos(phi2) * t_1) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)))) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	tmp = 0
	if (t_0 <= -0.018) or not (t_0 <= 0.07):
		tmp = R * (2.0 * math.atan2(math.sqrt(((math.cos(phi1) * t_1) + math.pow(math.sin((phi1 * 0.5)), 2.0))), math.sqrt((1.0 - ((math.cos(phi2) * t_1) + math.pow(math.sin((phi2 * -0.5)), 2.0))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((1.0 - math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	tmp = 0.0
	if ((t_0 <= -0.018) || !(t_0 <= 0.07))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi1) * t_1) + (sin(Float64(phi1 * 0.5)) ^ 2.0))), sqrt(Float64(1.0 - Float64(Float64(cos(phi2) * t_1) + (sin(Float64(phi2 * -0.5)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2)))) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	tmp = 0.0;
	if ((t_0 <= -0.018) || ~((t_0 <= 0.07)))
		tmp = R * (2.0 * atan2(sqrt(((cos(phi1) * t_1) + (sin((phi1 * 0.5)) ^ 2.0))), sqrt((1.0 - ((cos(phi2) * t_1) + (sin((phi2 * -0.5)) ^ 2.0))))));
	else
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((1.0 - (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.018], N[Not[LessEqual[t$95$0, 0.07]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 11: 52.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{-6} \lor \neg \left(t_1 \leq 0.02\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0}}{\sqrt{1 - t_0}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 \cdot \left(t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (+
          (* (cos phi2) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
          (pow (sin (* phi2 -0.5)) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= t_1 -5e-6) (not (<= t_1 0.02)))
     (* R (* 2.0 (atan2 (sqrt t_0) (sqrt (- 1.0 t_0)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_1 (* t_1 (* (cos phi1) (cos phi2))))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (cos(phi2) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)) + pow(sin((phi2 * -0.5)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((t_1 <= -5e-6) || !(t_1 <= 0.02)) {
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(((t_1 * (t_1 * (cos(phi1) * cos(phi2)))) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)) + (sin((phi2 * (-0.5d0))) ** 2.0d0)
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    if ((t_1 <= (-5d-6)) .or. (.not. (t_1 <= 0.02d0))) then
        tmp = r * (2.0d0 * atan2(sqrt(t_0), sqrt((1.0d0 - t_0))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((t_1 * (t_1 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (Math.cos(phi2) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)) + Math.pow(Math.sin((phi2 * -0.5)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((t_1 <= -5e-6) || !(t_1 <= 0.02)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_0), Math.sqrt((1.0 - t_0))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_1 * (t_1 * (Math.cos(phi1) * Math.cos(phi2)))) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (math.cos(phi2) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)) + math.pow(math.sin((phi2 * -0.5)), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if (t_1 <= -5e-6) or not (t_1 <= 0.02):
		tmp = R * (2.0 * math.atan2(math.sqrt(t_0), math.sqrt((1.0 - t_0))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_1 * (t_1 * (math.cos(phi1) * math.cos(phi2)))) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((1.0 - math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(cos(phi2) * (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((t_1 <= -5e-6) || !(t_1 <= 0.02))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_0), sqrt(Float64(1.0 - t_0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_1 * Float64(t_1 * Float64(cos(phi1) * cos(phi2)))) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (cos(phi2) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)) + (sin((phi2 * -0.5)) ^ 2.0);
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if ((t_1 <= -5e-6) || ~((t_1 <= 0.02)))
		tmp = R * (2.0 * atan2(sqrt(t_0), sqrt((1.0 - t_0))));
	else
		tmp = R * (2.0 * atan2(sqrt(((t_1 * (t_1 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((1.0 - (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$1, -5e-6], N[Not[LessEqual[t$95$1, 0.02]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$0], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$0), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$1 * N[(t$95$1 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 12: 59.5% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 13: 61.9% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 14: 43.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ t_3 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_4 := \sqrt{t_1 \cdot t_2 + t_3}\\ t_5 := \sqrt{1 - {t_0}^{2}}\\ \mathbf{if}\;t_1 \leq -0.018:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{t_5}\right)\\ \mathbf{elif}\;t_1 \leq 0.09:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 + t_2 \cdot \left|t_0\right|}}{t_5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* (- lambda1 lambda2) 0.5)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* t_1 (* (cos phi1) (cos phi2))))
        (t_3 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_4 (sqrt (+ (* t_1 t_2) t_3)))
        (t_5 (sqrt (- 1.0 (pow t_0 2.0)))))
   (if (<= t_1 -0.018)
     (* R (* 2.0 (atan2 t_4 t_5)))
     (if (<= t_1 0.09)
       (*
        R
        (*
         2.0
         (atan2 t_4 (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))))
       (* R (* 2.0 (atan2 (sqrt (+ t_3 (* t_2 (fabs t_0)))) t_5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) * 0.5));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = t_1 * (cos(phi1) * cos(phi2));
	double t_3 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = sqrt(((t_1 * t_2) + t_3));
	double t_5 = sqrt((1.0 - pow(t_0, 2.0)));
	double tmp;
	if (t_1 <= -0.018) {
		tmp = R * (2.0 * atan2(t_4, t_5));
	} else if (t_1 <= 0.09) {
		tmp = R * (2.0 * atan2(t_4, sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_3 + (t_2 * fabs(t_0)))), t_5));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) * 0.5d0))
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = t_1 * (cos(phi1) * cos(phi2))
    t_3 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_4 = sqrt(((t_1 * t_2) + t_3))
    t_5 = sqrt((1.0d0 - (t_0 ** 2.0d0)))
    if (t_1 <= (-0.018d0)) then
        tmp = r * (2.0d0 * atan2(t_4, t_5))
    else if (t_1 <= 0.09d0) then
        tmp = r * (2.0d0 * atan2(t_4, sqrt((1.0d0 - (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_3 + (t_2 * abs(t_0)))), t_5))
    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) * 0.5));
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = t_1 * (Math.cos(phi1) * Math.cos(phi2));
	double t_3 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_4 = Math.sqrt(((t_1 * t_2) + t_3));
	double t_5 = Math.sqrt((1.0 - Math.pow(t_0, 2.0)));
	double tmp;
	if (t_1 <= -0.018) {
		tmp = R * (2.0 * Math.atan2(t_4, t_5));
	} else if (t_1 <= 0.09) {
		tmp = R * (2.0 * Math.atan2(t_4, Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_3 + (t_2 * Math.abs(t_0)))), t_5));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) * 0.5))
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = t_1 * (math.cos(phi1) * math.cos(phi2))
	t_3 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_4 = math.sqrt(((t_1 * t_2) + t_3))
	t_5 = math.sqrt((1.0 - math.pow(t_0, 2.0)))
	tmp = 0
	if t_1 <= -0.018:
		tmp = R * (2.0 * math.atan2(t_4, t_5))
	elif t_1 <= 0.09:
		tmp = R * (2.0 * math.atan2(t_4, math.sqrt((1.0 - math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_3 + (t_2 * math.fabs(t_0)))), t_5))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) * 0.5))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(t_1 * Float64(cos(phi1) * cos(phi2)))
	t_3 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_4 = sqrt(Float64(Float64(t_1 * t_2) + t_3))
	t_5 = sqrt(Float64(1.0 - (t_0 ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -0.018)
		tmp = Float64(R * Float64(2.0 * atan(t_4, t_5)));
	elseif (t_1 <= 0.09)
		tmp = Float64(R * Float64(2.0 * atan(t_4, sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_3 + Float64(t_2 * abs(t_0)))), t_5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) * 0.5));
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = t_1 * (cos(phi1) * cos(phi2));
	t_3 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_4 = sqrt(((t_1 * t_2) + t_3));
	t_5 = sqrt((1.0 - (t_0 ^ 2.0)));
	tmp = 0.0;
	if (t_1 <= -0.018)
		tmp = R * (2.0 * atan2(t_4, t_5));
	elseif (t_1 <= 0.09)
		tmp = R * (2.0 * atan2(t_4, sqrt((1.0 - (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_3 + (t_2 * abs(t_0)))), t_5));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(N[(t$95$1 * t$95$2), $MachinePrecision] + t$95$3), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.018], N[(R * N[(2.0 * N[ArcTan[t$95$4 / t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.09], N[(R * N[(2.0 * N[ArcTan[t$95$4 / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$3 + N[(t$95$2 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 15: 62.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \cos \phi_2 \cdot t_0 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -8.5 \cdot 10^{-7} \lor \neg \left(\phi_2 \leq 1.75 \cdot 10^{-7}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1}}{\sqrt{1 - t_1}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2 \cdot \left(t_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{1 - \left(\cos \phi_1 \cdot t_0 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_1 (+ (* (cos phi2) t_0) (pow (sin (* phi2 -0.5)) 2.0)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= phi2 -8.5e-7) (not (<= phi2 1.75e-7)))
     (* R (* 2.0 (atan2 (sqrt t_1) (sqrt (- 1.0 t_1)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_2 (* t_2 (* (cos phi1) (cos phi2))))
          (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
        (sqrt
         (- 1.0 (+ (* (cos phi1) t_0) (pow (sin (* phi1 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 = (cos(phi2) * t_0) + pow(sin((phi2 * -0.5)), 2.0);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -8.5e-7) || !(phi2 <= 1.75e-7)) {
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(((t_2 * (t_2 * (cos(phi1) * cos(phi2)))) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt((1.0 - ((cos(phi1) * t_0) + pow(sin((phi1 * 0.5)), 2.0))))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    t_1 = (cos(phi2) * t_0) + (sin((phi2 * (-0.5d0))) ** 2.0d0)
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    if ((phi2 <= (-8.5d-7)) .or. (.not. (phi2 <= 1.75d-7))) then
        tmp = r * (2.0d0 * atan2(sqrt(t_1), sqrt((1.0d0 - t_1))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((t_2 * (t_2 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((1.0d0 - ((cos(phi1) * t_0) + (sin((phi1 * 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 t_1 = (Math.cos(phi2) * t_0) + Math.pow(Math.sin((phi2 * -0.5)), 2.0);
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -8.5e-7) || !(phi2 <= 1.75e-7)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_1), Math.sqrt((1.0 - t_1))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_2 * (t_2 * (Math.cos(phi1) * Math.cos(phi2)))) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt((1.0 - ((Math.cos(phi1) * t_0) + Math.pow(Math.sin((phi1 * 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)
	t_1 = (math.cos(phi2) * t_0) + math.pow(math.sin((phi2 * -0.5)), 2.0)
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if (phi2 <= -8.5e-7) or not (phi2 <= 1.75e-7):
		tmp = R * (2.0 * math.atan2(math.sqrt(t_1), math.sqrt((1.0 - t_1))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_2 * (t_2 * (math.cos(phi1) * math.cos(phi2)))) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt((1.0 - ((math.cos(phi1) * t_0) + math.pow(math.sin((phi1 * 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 = Float64(Float64(cos(phi2) * t_0) + (sin(Float64(phi2 * -0.5)) ^ 2.0))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((phi2 <= -8.5e-7) || !(phi2 <= 1.75e-7))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_1), sqrt(Float64(1.0 - t_1)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_2 * Float64(t_2 * Float64(cos(phi1) * cos(phi2)))) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * t_0) + (sin(Float64(phi1 * 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;
	t_1 = (cos(phi2) * t_0) + (sin((phi2 * -0.5)) ^ 2.0);
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if ((phi2 <= -8.5e-7) || ~((phi2 <= 1.75e-7)))
		tmp = R * (2.0 * atan2(sqrt(t_1), sqrt((1.0 - t_1))));
	else
		tmp = R * (2.0 * atan2(sqrt(((t_2 * (t_2 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((1.0 - ((cos(phi1) * t_0) + (sin((phi1 * 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]}, Block[{t$95$1 = N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -8.5e-7], N[Not[LessEqual[phi2, 1.75e-7]], $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], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$2 * N[(t$95$2 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$0), $MachinePrecision] + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 16: 61.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\\ t_1 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_2 := \cos \phi_2 \cdot t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\phi_2 \leq -1.96 \cdot 10^{-16} \lor \neg \left(\phi_2 \leq 2 \cdot 10^{-8}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2}}{\sqrt{1 - t_2}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_3 \cdot \left(t_3 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + t_0}}{\sqrt{1 - \left(\cos \phi_1 \cdot t_1 + t_0\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* phi1 0.5)) 2.0))
        (t_1 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_2 (+ (* (cos phi2) t_1) (pow (sin (* phi2 -0.5)) 2.0)))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= phi2 -1.96e-16) (not (<= phi2 2e-8)))
     (* R (* 2.0 (atan2 (sqrt t_2) (sqrt (- 1.0 t_2)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (* t_3 (* t_3 (* (cos phi1) (cos phi2)))) t_0))
        (sqrt (- 1.0 (+ (* (cos phi1) t_1) t_0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin((phi1 * 0.5)), 2.0);
	double t_1 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = (cos(phi2) * t_1) + pow(sin((phi2 * -0.5)), 2.0);
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -1.96e-16) || !(phi2 <= 2e-8)) {
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(((t_3 * (t_3 * (cos(phi1) * cos(phi2)))) + t_0)), sqrt((1.0 - ((cos(phi1) * t_1) + t_0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_0 = sin((phi1 * 0.5d0)) ** 2.0d0
    t_1 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    t_2 = (cos(phi2) * t_1) + (sin((phi2 * (-0.5d0))) ** 2.0d0)
    t_3 = sin(((lambda1 - lambda2) / 2.0d0))
    if ((phi2 <= (-1.96d-16)) .or. (.not. (phi2 <= 2d-8))) then
        tmp = r * (2.0d0 * atan2(sqrt(t_2), sqrt((1.0d0 - t_2))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((t_3 * (t_3 * (cos(phi1) * cos(phi2)))) + t_0)), sqrt((1.0d0 - ((cos(phi1) * t_1) + t_0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin((phi1 * 0.5)), 2.0);
	double t_1 = Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = (Math.cos(phi2) * t_1) + Math.pow(Math.sin((phi2 * -0.5)), 2.0);
	double t_3 = Math.sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((phi2 <= -1.96e-16) || !(phi2 <= 2e-8)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_2), Math.sqrt((1.0 - t_2))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_3 * (t_3 * (Math.cos(phi1) * Math.cos(phi2)))) + t_0)), Math.sqrt((1.0 - ((Math.cos(phi1) * t_1) + t_0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin((phi1 * 0.5)), 2.0)
	t_1 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	t_2 = (math.cos(phi2) * t_1) + math.pow(math.sin((phi2 * -0.5)), 2.0)
	t_3 = math.sin(((lambda1 - lambda2) / 2.0))
	tmp = 0
	if (phi2 <= -1.96e-16) or not (phi2 <= 2e-8):
		tmp = R * (2.0 * math.atan2(math.sqrt(t_2), math.sqrt((1.0 - t_2))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_3 * (t_3 * (math.cos(phi1) * math.cos(phi2)))) + t_0)), math.sqrt((1.0 - ((math.cos(phi1) * t_1) + t_0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(phi1 * 0.5)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_2 = Float64(Float64(cos(phi2) * t_1) + (sin(Float64(phi2 * -0.5)) ^ 2.0))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((phi2 <= -1.96e-16) || !(phi2 <= 2e-8))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), sqrt(Float64(1.0 - t_2)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_3 * Float64(t_3 * Float64(cos(phi1) * cos(phi2)))) + t_0)), sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * t_1) + t_0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((phi1 * 0.5)) ^ 2.0;
	t_1 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	t_2 = (cos(phi2) * t_1) + (sin((phi2 * -0.5)) ^ 2.0);
	t_3 = sin(((lambda1 - lambda2) / 2.0));
	tmp = 0.0;
	if ((phi2 <= -1.96e-16) || ~((phi2 <= 2e-8)))
		tmp = R * (2.0 * atan2(sqrt(t_2), sqrt((1.0 - t_2))));
	else
		tmp = R * (2.0 * atan2(sqrt(((t_3 * (t_3 * (cos(phi1) * cos(phi2)))) + t_0)), sqrt((1.0 - ((cos(phi1) * t_1) + t_0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(phi1 * 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[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[phi2, -1.96e-16], N[Not[LessEqual[phi2, 2e-8]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$3 * N[(t$95$3 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 17: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}\\ \mathbf{if}\;t_0 \leq -0.018 \lor \neg \left(t_0 \leq 0.09\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (+
           (* t_0 (* t_0 (* (cos phi1) (cos phi2))))
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))))
   (if (or (<= t_0 -0.018) (not (<= t_0 0.09)))
     (*
      R
      (*
       2.0
       (atan2 t_1 (sqrt (- 1.0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))))
     (*
      R
      (*
       2.0
       (atan2 t_1 (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + pow(sin(((phi1 - phi2) / 2.0)), 2.0)));
	double tmp;
	if ((t_0 <= -0.018) || !(t_0 <= 0.09)) {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = sin(((lambda1 - lambda2) / 2.0d0))
    t_1 = sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0)))
    if ((t_0 <= (-0.018d0)) .or. (.not. (t_0 <= 0.09d0))) then
        tmp = r * (2.0d0 * atan2(t_1, sqrt((1.0d0 - (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)))))
    else
        tmp = r * (2.0d0 * atan2(t_1, sqrt((1.0d0 - (sin((0.5d0 * (phi1 - phi2))) ** 2.0d0)))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.sqrt(((t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)))) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0)));
	double tmp;
	if ((t_0 <= -0.018) || !(t_0 <= 0.09)) {
		tmp = R * (2.0 * Math.atan2(t_1, Math.sqrt((1.0 - Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_1, Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.sqrt(((t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)))
	tmp = 0
	if (t_0 <= -0.018) or not (t_0 <= 0.09):
		tmp = R * (2.0 * math.atan2(t_1, math.sqrt((1.0 - math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))))
	else:
		tmp = R * (2.0 * math.atan2(t_1, math.sqrt((1.0 - math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0)))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sqrt(Float64(Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2)))) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0)))
	tmp = 0.0
	if ((t_0 <= -0.018) || !(t_0 <= 0.09))
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((lambda1 - lambda2) / 2.0));
	t_1 = sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0)));
	tmp = 0.0;
	if ((t_0 <= -0.018) || ~((t_0 <= 0.09)))
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)))));
	else
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - (sin((0.5 * (phi1 - phi2))) ^ 2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[(t$95$0 * N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.018], N[Not[LessEqual[t$95$0, 0.09]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[t$95$1 / 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[t$95$1 / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 18: 61.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_2 := \cos \phi_2 \cdot t_1 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\\ \mathbf{if}\;\phi_1 \leq -0.19 \lor \neg \left(\phi_1 \leq 1.95 \cdot 10^{-15}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + \left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot t_1 + {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_2}}{\sqrt{1 - t_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_2 (+ (* (cos phi2) t_1) (pow (sin (* phi2 -0.5)) 2.0))))
   (if (or (<= phi1 -0.19) (not (<= phi1 1.95e-15)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* t_0 (* t_0 (* (cos phi1) (cos phi2))))
          (- 0.5 (/ (cos (- phi1 phi2)) 2.0))))
        (sqrt (- 1.0 (+ (* (cos phi1) t_1) (pow (sin (* phi1 0.5)) 2.0)))))))
     (* R (* 2.0 (atan2 (sqrt t_2) (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 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = (cos(phi2) * t_1) + pow(sin((phi2 * -0.5)), 2.0);
	double tmp;
	if ((phi1 <= -0.19) || !(phi1 <= 1.95e-15)) {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt((1.0 - ((cos(phi1) * t_1) + pow(sin((phi1 * 0.5)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(t_2), 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 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    t_2 = (cos(phi2) * t_1) + (sin((phi2 * (-0.5d0))) ** 2.0d0)
    if ((phi1 <= (-0.19d0)) .or. (.not. (phi1 <= 1.95d-15))) then
        tmp = r * (2.0d0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (0.5d0 - (cos((phi1 - phi2)) / 2.0d0)))), sqrt((1.0d0 - ((cos(phi1) * t_1) + (sin((phi1 * 0.5d0)) ** 2.0d0))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(t_2), 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.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_2 = (Math.cos(phi2) * t_1) + Math.pow(Math.sin((phi2 * -0.5)), 2.0);
	double tmp;
	if ((phi1 <= -0.19) || !(phi1 <= 1.95e-15)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)))) + (0.5 - (Math.cos((phi1 - phi2)) / 2.0)))), Math.sqrt((1.0 - ((Math.cos(phi1) * t_1) + Math.pow(Math.sin((phi1 * 0.5)), 2.0))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(t_2), 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.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	t_2 = (math.cos(phi2) * t_1) + math.pow(math.sin((phi2 * -0.5)), 2.0)
	tmp = 0
	if (phi1 <= -0.19) or not (phi1 <= 1.95e-15):
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))) + (0.5 - (math.cos((phi1 - phi2)) / 2.0)))), math.sqrt((1.0 - ((math.cos(phi1) * t_1) + math.pow(math.sin((phi1 * 0.5)), 2.0))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt(t_2), 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 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_2 = Float64(Float64(cos(phi2) * t_1) + (sin(Float64(phi2 * -0.5)) ^ 2.0))
	tmp = 0.0
	if ((phi1 <= -0.19) || !(phi1 <= 1.95e-15))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2)))) + Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0)))), sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * t_1) + (sin(Float64(phi1 * 0.5)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(t_2), 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 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	t_2 = (cos(phi2) * t_1) + (sin((phi2 * -0.5)) ^ 2.0);
	tmp = 0.0;
	if ((phi1 <= -0.19) || ~((phi1 <= 1.95e-15)))
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (0.5 - (cos((phi1 - phi2)) / 2.0)))), sqrt((1.0 - ((cos(phi1) * t_1) + (sin((phi1 * 0.5)) ^ 2.0))))));
	else
		tmp = R * (2.0 * atan2(sqrt(t_2), 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[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[phi1, -0.19], N[Not[LessEqual[phi1, 1.95e-15]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi1], $MachinePrecision] * t$95$1), $MachinePrecision] + N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[t$95$2], $MachinePrecision] / N[Sqrt[N[(1.0 - t$95$2), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 19: 34.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\cos \left(\lambda_2 \cdot -0.5\right)}^{2}}\\ \mathbf{if}\;\lambda_2 \leq -7.8 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\cos \phi_2 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2} + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}}}{t_1}\right)\\ \mathbf{elif}\;\lambda_2 \leq 0.00175:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right) + {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}}}{\sqrt{{\cos \left(\lambda_1 \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2} + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right)}}{t_1}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (sqrt (pow (cos (* lambda2 -0.5)) 2.0))))
   (if (<= lambda2 -7.8e-5)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (+
          (* (cos phi2) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
          (pow (sin (* phi2 -0.5)) 2.0)))
        t_1)))
     (if (<= lambda2 0.00175)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (* t_0 (* t_0 (* (cos phi1) (cos phi2))))
            (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)))
          (sqrt (pow (cos (* lambda1 0.5)) 2.0)))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt
           (+
            (pow (sin (* 0.5 (- phi1 phi2))) 2.0)
            (* (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 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sqrt(pow(cos((lambda2 * -0.5)), 2.0));
	double tmp;
	if (lambda2 <= -7.8e-5) {
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)) + pow(sin((phi2 * -0.5)), 2.0))), t_1));
	} else if (lambda2 <= 0.00175) {
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + pow(sin(((phi1 - phi2) / 2.0)), 2.0))), sqrt(pow(cos((lambda1 * 0.5)), 2.0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin((0.5 * (phi1 - phi2))), 2.0) + (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(((lambda1 - lambda2) / 2.0d0))
    t_1 = sqrt((cos((lambda2 * (-0.5d0))) ** 2.0d0))
    if (lambda2 <= (-7.8d-5)) then
        tmp = r * (2.0d0 * atan2(sqrt(((cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)) + (sin((phi2 * (-0.5d0))) ** 2.0d0))), t_1))
    else if (lambda2 <= 0.00175d0) then
        tmp = r * (2.0d0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0))), sqrt((cos((lambda1 * 0.5d0)) ** 2.0d0))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin((0.5d0 * (phi1 - phi2))) ** 2.0d0) + (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.sin(((lambda1 - lambda2) / 2.0));
	double t_1 = Math.sqrt(Math.pow(Math.cos((lambda2 * -0.5)), 2.0));
	double tmp;
	if (lambda2 <= -7.8e-5) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((Math.cos(phi2) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))), t_1));
	} else if (lambda2 <= 0.00175) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)))) + Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0))), Math.sqrt(Math.pow(Math.cos((lambda1 * 0.5)), 2.0))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0) + (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.sin(((lambda1 - lambda2) / 2.0))
	t_1 = math.sqrt(math.pow(math.cos((lambda2 * -0.5)), 2.0))
	tmp = 0
	if lambda2 <= -7.8e-5:
		tmp = R * (2.0 * math.atan2(math.sqrt(((math.cos(phi2) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)) + math.pow(math.sin((phi2 * -0.5)), 2.0))), t_1))
	elif lambda2 <= 0.00175:
		tmp = R * (2.0 * math.atan2(math.sqrt(((t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))) + math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0))), math.sqrt(math.pow(math.cos((lambda1 * 0.5)), 2.0))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin((0.5 * (phi1 - phi2))), 2.0) + (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(lambda1 - lambda2) / 2.0))
	t_1 = sqrt((cos(Float64(lambda2 * -0.5)) ^ 2.0))
	tmp = 0.0
	if (lambda2 <= -7.8e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(cos(phi2) * (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0)) + (sin(Float64(phi2 * -0.5)) ^ 2.0))), t_1)));
	elseif (lambda2 <= 0.00175)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2)))) + (sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((cos(Float64(lambda1 * 0.5)) ^ 2.0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0) + Float64(cos(phi1) * Float64(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(((lambda1 - lambda2) / 2.0));
	t_1 = sqrt((cos((lambda2 * -0.5)) ^ 2.0));
	tmp = 0.0;
	if (lambda2 <= -7.8e-5)
		tmp = R * (2.0 * atan2(sqrt(((cos(phi2) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)) + (sin((phi2 * -0.5)) ^ 2.0))), t_1));
	elseif (lambda2 <= 0.00175)
		tmp = R * (2.0 * atan2(sqrt(((t_0 * (t_0 * (cos(phi1) * cos(phi2)))) + (sin(((phi1 - phi2) / 2.0)) ^ 2.0))), sqrt((cos((lambda1 * 0.5)) ^ 2.0))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin((0.5 * (phi1 - phi2))) ^ 2.0) + (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[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[Power[N[Cos[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[lambda2, -7.8e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda2, 0.00175], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(t$95$0 * N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Power[N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(0.5 * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(lambda2 * -0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 20: 34.6% accurate, 1.4× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 21: 30.0% accurate, 1.4× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 22: 27.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(((cos(phi1) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)) + pow(sin((phi1 * 0.5)), 2.0))), sqrt(pow(cos((lambda2 * -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(((cos(phi1) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)) + (sin((phi1 * 0.5d0)) ** 2.0d0))), sqrt((cos((lambda2 * (-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.cos(phi1) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)) + Math.pow(Math.sin((phi1 * 0.5)), 2.0))), Math.sqrt(Math.pow(Math.cos((lambda2 * -0.5)), 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Alternative 23: 28.0% accurate, 1.5× speedup

Math

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

FPCore

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

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * (2.0 * atan2(sqrt(((cos(phi2) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)) + pow(sin((phi2 * -0.5)), 2.0))), sqrt(pow(cos((lambda2 * -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(((cos(phi2) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)) + (sin((phi2 * (-0.5d0))) ** 2.0d0))), sqrt((cos((lambda2 * (-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.cos(phi2) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))), Math.sqrt(Math.pow(Math.cos((lambda2 * -0.5)), 2.0))));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

&prev;&pcontext;&pcontext2;&ctx;
1. Add Preprocessing

Reproduce

herbie shell --seed 2023364 
(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))))))))))