Distance on a great circle

Percentage Accurate: 62.2% → 78.3%

Time: 2.2min

Alternatives: 25

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_0 \cdot t_0\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(-\cos \left(\frac{\phi_1}{2}\right)\right)\right)\right)}^{2}\right)}\\ t_2 := {\left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}^{2}\\ \mathbf{if}\;\lambda_2 \leq -4.1 \cdot 10^{-5} \lor \neg \left(\lambda_2 \leq 1.05 \cdot 10^{-5}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right) + t_2\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - \left(t_2 + \cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\lambda_1 \cdot 0.5\right)}^{2}\right)\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (fma
           (cos phi1)
           (* (cos phi2) (* t_0 t_0))
           (pow
            (fma
             (sin (/ phi1 2.0))
             (cos (/ phi2 2.0))
             (* (sin (/ phi2 2.0)) (- (cos (/ phi1 2.0)))))
            2.0))))
        (t_2
         (pow
          (-
           (* (sin (* phi2 0.5)) (cos (* phi1 0.5)))
           (* (cos (* phi2 0.5)) (sin (* phi1 0.5))))
          2.0)))
   (if (or (<= lambda2 -4.1e-5) (not (<= lambda2 1.05e-5)))
     (*
      R
      (*
       2.0
       (atan2
        t_1
        (sqrt
         (-
          1.0
          (+
           (* (cos phi1) (* (cos phi2) (pow (sin (* lambda2 -0.5)) 2.0)))
           t_2))))))
     (*
      R
      (*
       2.0
       (atan2
        t_1
        (sqrt
         (-
          1.0
          (+
           t_2
           (*
            (cos phi1)
            (* (cos phi2) (pow (sin (* lambda1 0.5)) 2.0))))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin(((lambda1 - lambda2) / 2.0));
	double t_1 = sqrt(fma(cos(phi1), (cos(phi2) * (t_0 * t_0)), pow(fma(sin((phi1 / 2.0)), cos((phi2 / 2.0)), (sin((phi2 / 2.0)) * -cos((phi1 / 2.0)))), 2.0)));
	double t_2 = pow(((sin((phi2 * 0.5)) * cos((phi1 * 0.5))) - (cos((phi2 * 0.5)) * sin((phi1 * 0.5)))), 2.0);
	double tmp;
	if ((lambda2 <= -4.1e-5) || !(lambda2 <= 1.05e-5)) {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - ((cos(phi1) * (cos(phi2) * pow(sin((lambda2 * -0.5)), 2.0))) + t_2)))));
	} else {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - (t_2 + (cos(phi1) * (cos(phi2) * pow(sin((lambda1 * 0.5)), 2.0))))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_0 * t_0)), (fma(sin(Float64(phi1 / 2.0)), cos(Float64(phi2 / 2.0)), Float64(sin(Float64(phi2 / 2.0)) * Float64(-cos(Float64(phi1 / 2.0))))) ^ 2.0)))
	t_2 = Float64(Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(phi1 * 0.5))) - Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(phi1 * 0.5)))) ^ 2.0
	tmp = 0.0
	if ((lambda2 <= -4.1e-5) || !(lambda2 <= 1.05e-5))
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * Float64(cos(phi2) * (sin(Float64(lambda2 * -0.5)) ^ 2.0))) + t_2))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - Float64(t_2 + Float64(cos(phi1) * Float64(cos(phi2) * (sin(Float64(lambda1 * 0.5)) ^ 2.0)))))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[(N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]}, If[Or[LessEqual[lambda2, -4.1e-5], N[Not[LessEqual[lambda2, 1.05e-5]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(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] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(t$95$2 + N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

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

Alternative 2: 78.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \left(\frac{\phi_1}{2}\right)\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \left(\frac{\phi_2}{2}\right)\\ t_3 := \sin \left(\frac{\phi_2}{2}\right)\\ t_4 := \sin \left(\frac{\phi_1}{2}\right)\\ t_5 := \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_1 \cdot t_1\right), {\left(\mathsf{fma}\left(t_4, t_2, t_3 \cdot \left(-t_0\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \mathsf{fma}\left(\cos \phi_2, \cos \phi_1 \cdot \left(\log \left(e^{t_5}\right) \cdot \sqrt[3]{{t_5}^{3}}\right), {\left(t_0 \cdot t_3 - t_4 \cdot t_2\right)}^{2}\right)}}\right) \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (/ phi1 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (cos (/ phi2 2.0)))
        (t_3 (sin (/ phi2 2.0)))
        (t_4 (sin (/ phi1 2.0)))
        (t_5 (sin (* (- lambda1 lambda2) 0.5))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (fma
        (cos phi1)
        (* (cos phi2) (* t_1 t_1))
        (pow (fma t_4 t_2 (* t_3 (- t_0))) 2.0)))
      (sqrt
       (-
        1.0
        (fma
         (cos phi2)
         (* (cos phi1) (* (log (exp t_5)) (cbrt (pow t_5 3.0))))
         (pow (- (* t_0 t_3) (* t_4 t_2)) 2.0)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi1 / 2.0));
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos((phi2 / 2.0));
	double t_3 = sin((phi2 / 2.0));
	double t_4 = sin((phi1 / 2.0));
	double t_5 = sin(((lambda1 - lambda2) * 0.5));
	return R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (t_1 * t_1)), pow(fma(t_4, t_2, (t_3 * -t_0)), 2.0))), sqrt((1.0 - fma(cos(phi2), (cos(phi1) * (log(exp(t_5)) * cbrt(pow(t_5, 3.0)))), pow(((t_0 * t_3) - (t_4 * t_2)), 2.0))))));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi1 / 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = cos(Float64(phi2 / 2.0))
	t_3 = sin(Float64(phi2 / 2.0))
	t_4 = sin(Float64(phi1 / 2.0))
	t_5 = sin(Float64(Float64(lambda1 - lambda2) * 0.5))
	return Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_1 * t_1)), (fma(t_4, t_2, Float64(t_3 * Float64(-t_0))) ^ 2.0))), sqrt(Float64(1.0 - fma(cos(phi2), Float64(cos(phi1) * Float64(log(exp(t_5)) * cbrt((t_5 ^ 3.0)))), (Float64(Float64(t_0 * t_3) - Float64(t_4 * t_2)) ^ 2.0)))))))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$4 * t$95$2 + N[(t$95$3 * (-t$95$0)), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Cos[phi2], $MachinePrecision] * N[(N[Cos[phi1], $MachinePrecision] * N[(N[Log[N[Exp[t$95$5], $MachinePrecision]], $MachinePrecision] * N[Power[N[Power[t$95$5, 3.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(t$95$0 * t$95$3), $MachinePrecision] - N[(t$95$4 * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

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

Alternative 3: 78.4% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

Alternative 4: 78.4% accurate, 0.7× speedup

Math

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

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

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

Alternative 5: 73.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \cos \phi_1 \cdot \cos \phi_2\\ t_1 := \cos \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\\ t_2 := \sin \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\\ t_3 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ \mathbf{if}\;\lambda_1 \leq -1.55 \cdot 10^{-7} \lor \neg \left(\lambda_1 \leq 6 \cdot 10^{-12}\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_3 \cdot \left(t_3 \cdot t_0\right)}}{\sqrt{e^{\mathsf{log1p}\left(-\mathsf{fma}\left(t_0, 0.5 + -0.5 \cdot \left(\cos \lambda_1 \cdot \cos \lambda_2 + \sin \lambda_1 \cdot \sin \lambda_2\right), {\left(t_1 - t_2\right)}^{2}\right)\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_3 \cdot t_3\right), {\left(\mathsf{fma}\left(\sin \left(\frac{\phi_1}{2}\right), \cos \left(\frac{\phi_2}{2}\right), \sin \left(\frac{\phi_2}{2}\right) \cdot \left(-\cos \left(\frac{\phi_1}{2}\right)\right)\right)\right)}^{2}\right)}}{\sqrt{1 - \left(\cos \phi_1 \cdot \left(\cos \phi_2 \cdot {\sin \left(\lambda_2 \cdot -0.5\right)}^{2}\right) + {\left(t_2 - t_1\right)}^{2}\right)}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (cos phi1) (cos phi2)))
        (t_1 (* (cos (* phi2 0.5)) (sin (* phi1 0.5))))
        (t_2 (* (sin (* phi2 0.5)) (cos (* phi1 0.5))))
        (t_3 (sin (/ (- lambda1 lambda2) 2.0))))
   (if (or (<= lambda1 -1.55e-7) (not (<= lambda1 6e-12)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) (* t_3 (* t_3 t_0))))
        (sqrt
         (exp
          (log1p
           (-
            (fma
             t_0
             (+
              0.5
              (*
               -0.5
               (+
                (* (cos lambda1) (cos lambda2))
                (* (sin lambda1) (sin lambda2)))))
             (pow (- t_1 t_2) 2.0)))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt
         (fma
          (cos phi1)
          (* (cos phi2) (* t_3 t_3))
          (pow
           (fma
            (sin (/ phi1 2.0))
            (cos (/ phi2 2.0))
            (* (sin (/ phi2 2.0)) (- (cos (/ phi1 2.0)))))
           2.0)))
        (sqrt
         (-
          1.0
          (+
           (* (cos phi1) (* (cos phi2) (pow (sin (* lambda2 -0.5)) 2.0)))
           (pow (- t_2 t_1) 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 = cos((phi2 * 0.5)) * sin((phi1 * 0.5));
	double t_2 = sin((phi2 * 0.5)) * cos((phi1 * 0.5));
	double t_3 = sin(((lambda1 - lambda2) / 2.0));
	double tmp;
	if ((lambda1 <= -1.55e-7) || !(lambda1 <= 6e-12)) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_3 * (t_3 * t_0)))), sqrt(exp(log1p(-fma(t_0, (0.5 + (-0.5 * ((cos(lambda1) * cos(lambda2)) + (sin(lambda1) * sin(lambda2))))), pow((t_1 - t_2), 2.0)))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (t_3 * t_3)), pow(fma(sin((phi1 / 2.0)), cos((phi2 / 2.0)), (sin((phi2 / 2.0)) * -cos((phi1 / 2.0)))), 2.0))), sqrt((1.0 - ((cos(phi1) * (cos(phi2) * pow(sin((lambda2 * -0.5)), 2.0))) + pow((t_2 - t_1), 2.0))))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(cos(phi1) * cos(phi2))
	t_1 = Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(phi1 * 0.5)))
	t_2 = Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(phi1 * 0.5)))
	t_3 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	tmp = 0.0
	if ((lambda1 <= -1.55e-7) || !(lambda1 <= 6e-12))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_3 * Float64(t_3 * t_0)))), sqrt(exp(log1p(Float64(-fma(t_0, Float64(0.5 + Float64(-0.5 * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))), (Float64(t_1 - t_2) ^ 2.0)))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_3 * t_3)), (fma(sin(Float64(phi1 / 2.0)), cos(Float64(phi2 / 2.0)), Float64(sin(Float64(phi2 / 2.0)) * Float64(-cos(Float64(phi1 / 2.0))))) ^ 2.0))), sqrt(Float64(1.0 - Float64(Float64(cos(phi1) * Float64(cos(phi2) * (sin(Float64(lambda2 * -0.5)) ^ 2.0))) + (Float64(t_2 - t_1) ^ 2.0)))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, If[Or[LessEqual[lambda1, -1.55e-7], N[Not[LessEqual[lambda1, 6e-12]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$3 * N[(t$95$3 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Exp[N[Log[1 + (-N[(t$95$0 * N[(0.5 + N[(-0.5 * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(t$95$1 - t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $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[(t$95$3 * t$95$3), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[Sin[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] + N[(N[Sin[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision] * (-N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(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] + N[Power[N[(t$95$2 - t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

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

Alternative 6: 63.6% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_1 = Float64(cos(phi1) * cos(phi2))
	return Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_0 * Float64(t_0 * t_1)))), sqrt(exp(log1p(Float64(-fma(t_1, Float64(0.5 + Float64(-0.5 * Float64(Float64(cos(lambda1) * cos(lambda2)) + Float64(sin(lambda1) * sin(lambda2))))), (Float64(Float64(cos(Float64(phi2 * 0.5)) * sin(Float64(phi1 * 0.5))) - Float64(sin(Float64(phi2 * 0.5)) * cos(Float64(phi1 * 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]}, Block[{t$95$1 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$0 * N[(t$95$0 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Exp[N[Log[1 + (-N[(t$95$1 * N[(0.5 + N[(-0.5 * N[(N[(N[Cos[lambda1], $MachinePrecision] * N[Cos[lambda2], $MachinePrecision]), $MachinePrecision] + N[(N[Sin[lambda1], $MachinePrecision] * N[Sin[lambda2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 7: 63.1% 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: 63.1% accurate, 0.8× speedup

Math

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

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 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_1))
      (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((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), 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(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + t_1)), 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((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), 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((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), 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((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), 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(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), 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[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(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: 52.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{1 - \left({\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)}\\ t_1 := 0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t_2 \cdot \left(t_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;t_2 \leq -0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_3}}{t_0}\right)\\ \mathbf{elif}\;t_2 \leq 0.006:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_3}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\mathsf{fma}\left(\cos \phi_1, \cos \phi_2 \cdot \left(t_2 \cdot t_2\right), t_1\right)}}{t_0}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0
         (sqrt
          (-
           1.0
           (+
            (pow (sin (* phi1 0.5)) 2.0)
            (* (cos phi1) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))))
        (t_1 (- 0.5 (/ (cos (- phi1 phi2)) 2.0)))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* t_2 (* t_2 (* (cos phi1) (cos phi2))))))
   (if (<= t_2 -0.01)
     (* R (* 2.0 (atan2 (sqrt (+ t_1 t_3)) t_0)))
     (if (<= t_2 0.006)
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_3))
          (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))))
       (*
        R
        (*
         2.0
         (atan2
          (sqrt (fma (cos phi1) (* (cos phi2) (* t_2 t_2)) t_1))
          t_0)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sqrt((1.0 - (pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))));
	double t_1 = 0.5 - (cos((phi1 - phi2)) / 2.0);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_2 * (t_2 * (cos(phi1) * cos(phi2)));
	double tmp;
	if (t_2 <= -0.01) {
		tmp = R * (2.0 * atan2(sqrt((t_1 + t_3)), t_0));
	} else if (t_2 <= 0.006) {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_3)), sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(sqrt(fma(cos(phi1), (cos(phi2) * (t_2 * t_2)), t_1)), t_0));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sqrt(Float64(1.0 - Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0)))))
	t_1 = Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0))
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(t_2 * Float64(t_2 * Float64(cos(phi1) * cos(phi2))))
	tmp = 0.0
	if (t_2 <= -0.01)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_1 + t_3)), t_0)));
	elseif (t_2 <= 0.006)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_3)), sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(fma(cos(phi1), Float64(cos(phi2) * Float64(t_2 * t_2)), t_1)), t_0)));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(t$95$2 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -0.01], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$1 + t$95$3), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.006], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$3), $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], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Cos[phi1], $MachinePrecision] * N[(N[Cos[phi2], $MachinePrecision] * N[(t$95$2 * t$95$2), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

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

Alternative 10: 62.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

Alternative 11: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 0.006\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{\left(0.5 - \frac{\cos \left(\phi_1 - \phi_2\right)}{2}\right) + t_1}}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1}}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1 (* t_0 (* t_0 (* (cos phi1) (cos phi2))))))
   (if (or (<= t_0 -0.01) (not (<= t_0 0.006)))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (- 0.5 (/ (cos (- phi1 phi2)) 2.0)) t_1))
        (sqrt
         (-
          1.0
          (+
           (pow (sin (* phi1 0.5)) 2.0)
           (* (cos phi1) (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))))))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_1))
        (sqrt (- 1.0 (pow (sin (* 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 = t_0 * (t_0 * (cos(phi1) * cos(phi2)));
	double tmp;
	if ((t_0 <= -0.01) || !(t_0 <= 0.006)) {
		tmp = R * (2.0 * atan2(sqrt(((0.5 - (cos((phi1 - phi2)) / 2.0)) + t_1)), sqrt((1.0 - (pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * pow(sin(((lambda1 - lambda2) * 0.5)), 2.0)))))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), sqrt((1.0 - pow(sin((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 = t_0 * (t_0 * (cos(phi1) * cos(phi2)))
    if ((t_0 <= (-0.01d0)) .or. (.not. (t_0 <= 0.006d0))) then
        tmp = r * (2.0d0 * atan2(sqrt(((0.5d0 - (cos((phi1 - phi2)) / 2.0d0)) + t_1)), sqrt((1.0d0 - ((sin((phi1 * 0.5d0)) ** 2.0d0) + (cos(phi1) * (sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0)))))))
    else
        tmp = r * (2.0d0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + t_1)), sqrt((1.0d0 - (sin((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 = t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)));
	double tmp;
	if ((t_0 <= -0.01) || !(t_0 <= 0.006)) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt(((0.5 - (Math.cos((phi1 - phi2)) / 2.0)) + t_1)), Math.sqrt((1.0 - (Math.pow(Math.sin((phi1 * 0.5)), 2.0) + (Math.cos(phi1) * Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), Math.sqrt((1.0 - Math.pow(Math.sin((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 = t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))
	tmp = 0
	if (t_0 <= -0.01) or not (t_0 <= 0.006):
		tmp = R * (2.0 * math.atan2(math.sqrt(((0.5 - (math.cos((phi1 - phi2)) / 2.0)) + t_1)), math.sqrt((1.0 - (math.pow(math.sin((phi1 * 0.5)), 2.0) + (math.cos(phi1) * math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)))))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_1)), math.sqrt((1.0 - math.pow(math.sin((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 = Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2))))
	tmp = 0.0
	if ((t_0 <= -0.01) || !(t_0 <= 0.006))
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0)) + t_1)), sqrt(Float64(1.0 - Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), sqrt(Float64(1.0 - (sin(Float64(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 = t_0 * (t_0 * (cos(phi1) * cos(phi2)));
	tmp = 0.0;
	if ((t_0 <= -0.01) || ~((t_0 <= 0.006)))
		tmp = R * (2.0 * atan2(sqrt(((0.5 - (cos((phi1 - phi2)) / 2.0)) + t_1)), sqrt((1.0 - ((sin((phi1 * 0.5)) ^ 2.0) + (cos(phi1) * (sin(((lambda1 - lambda2) * 0.5)) ^ 2.0)))))));
	else
		tmp = R * (2.0 * atan2(sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + t_1)), sqrt((1.0 - (sin((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[(t$95$0 * N[(t$95$0 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.01], N[Not[LessEqual[t$95$0, 0.006]], $MachinePrecision]], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[Power[N[Sin[N[(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: 43.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\\ \mathbf{if}\;t_1 \leq -0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{t_0}}\right)\\ \mathbf{elif}\;t_1 \leq 0.0202:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\sqrt[3]{{t_0}^{3}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (sqrt
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* t_1 (* t_1 (* (cos phi1) (cos phi2))))))))
   (if (<= t_1 -0.01)
     (* R (* 2.0 (atan2 t_2 (sqrt t_0))))
     (if (<= t_1 0.0202)
       (*
        R
        (*
         2.0
         (atan2 t_2 (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))))
       (* R (* 2.0 (atan2 t_2 (sqrt (cbrt (pow t_0 3.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 1.0 - pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_1 * (cos(phi1) * cos(phi2))))));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = R * (2.0 * atan2(t_2, sqrt(t_0)));
	} else if (t_1 <= 0.0202) {
		tmp = R * (2.0 * atan2(t_2, sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(t_2, sqrt(cbrt(pow(t_0, 3.0)))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 1.0 - Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_1 * (Math.cos(phi1) * Math.cos(phi2))))));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = R * (2.0 * Math.atan2(t_2, Math.sqrt(t_0)));
	} else if (t_1 <= 0.0202) {
		tmp = R * (2.0 * Math.atan2(t_2, Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_2, Math.sqrt(Math.cbrt(Math.pow(t_0, 3.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(1.0 - (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_1 * Float64(t_1 * Float64(cos(phi1) * cos(phi2))))))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(t_0))));
	elseif (t_1 <= 0.0202)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(cbrt((t_0 ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(1.0 - N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$1 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[t$95$0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0202], N[(R * N[(2.0 * N[ArcTan[t$95$2 / 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[t$95$2 / N[Sqrt[N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

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

Alternative 13: 43.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_1 \cdot \left(t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\\ \mathbf{if}\;t_1 \leq -0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\log \left(1 + \mathsf{expm1}\left(t_0\right)\right)}}\right)\\ \mathbf{elif}\;t_1 \leq 0.0202:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_2}{\sqrt{\sqrt[3]{{t_0}^{3}}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (- 1.0 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0)))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2
         (sqrt
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* t_1 (* t_1 (* (cos phi1) (cos phi2))))))))
   (if (<= t_1 -0.01)
     (* R (* 2.0 (atan2 t_2 (sqrt (log (+ 1.0 (expm1 t_0)))))))
     (if (<= t_1 0.0202)
       (*
        R
        (*
         2.0
         (atan2 t_2 (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))))
       (* R (* 2.0 (atan2 t_2 (sqrt (cbrt (pow t_0 3.0))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 1.0 - pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_1 * (cos(phi1) * cos(phi2))))));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = R * (2.0 * atan2(t_2, sqrt(log((1.0 + expm1(t_0))))));
	} else if (t_1 <= 0.0202) {
		tmp = R * (2.0 * atan2(t_2, sqrt((1.0 - pow(sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * atan2(t_2, sqrt(cbrt(pow(t_0, 3.0)))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = 1.0 - Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_1 * (t_1 * (Math.cos(phi1) * Math.cos(phi2))))));
	double tmp;
	if (t_1 <= -0.01) {
		tmp = R * (2.0 * Math.atan2(t_2, Math.sqrt(Math.log((1.0 + Math.expm1(t_0))))));
	} else if (t_1 <= 0.0202) {
		tmp = R * (2.0 * Math.atan2(t_2, Math.sqrt((1.0 - Math.pow(Math.sin((0.5 * (phi1 - phi2))), 2.0)))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_2, Math.sqrt(Math.cbrt(Math.pow(t_0, 3.0)))));
	}
	return tmp;
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(1.0 - (sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0))
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_1 * Float64(t_1 * Float64(cos(phi1) * cos(phi2))))))
	tmp = 0.0
	if (t_1 <= -0.01)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(log(Float64(1.0 + expm1(t_0)))))));
	elseif (t_1 <= 0.0202)
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(Float64(1.0 - (sin(Float64(0.5 * Float64(phi1 - phi2))) ^ 2.0))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_2, sqrt(cbrt((t_0 ^ 3.0))))));
	end
	return tmp
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(1.0 - N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(t$95$1 * N[(t$95$1 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[t$95$1, -0.01], N[(R * N[(2.0 * N[ArcTan[t$95$2 / N[Sqrt[N[Log[N[(1.0 + N[(Exp[t$95$0] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0202], N[(R * N[(2.0 * N[ArcTan[t$95$2 / 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[t$95$2 / N[Sqrt[N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

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

Alternative 14: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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

Alternative 15: 59.6% accurate, 1.1× speedup

Math

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

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 (+ (- 0.5 (/ (cos (- phi1 phi2)) 2.0)) t_1))
      (sqrt (- 1.0 (+ (pow (sin (/ (- phi1 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(((0.5 - (cos((phi1 - phi2)) / 2.0)) + t_1)), sqrt((1.0 - (pow(sin(((phi1 - 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(((0.5d0 - (cos((phi1 - phi2)) / 2.0d0)) + t_1)), sqrt((1.0d0 - ((sin(((phi1 - 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(((0.5 - (Math.cos((phi1 - phi2)) / 2.0)) + t_1)), Math.sqrt((1.0 - (Math.pow(Math.sin(((phi1 - 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(((0.5 - (math.cos((phi1 - phi2)) / 2.0)) + t_1)), math.sqrt((1.0 - (math.pow(math.sin(((phi1 - 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(Float64(0.5 - Float64(cos(Float64(phi1 - phi2)) / 2.0)) + t_1)), sqrt(Float64(1.0 - Float64((sin(Float64(Float64(phi1 - 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(((0.5 - (cos((phi1 - phi2)) / 2.0)) + t_1)), sqrt((1.0 - ((sin(((phi1 - 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[(N[(0.5 - N[(N[Cos[N[(phi1 - phi2), $MachinePrecision]], $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

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

Alternative 16: 62.2% accurate, 1.1× speedup

Math

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

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 (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_1))
      (sqrt (+ 1.0 (- (- (/ (cos (- phi1 phi2)) 2.0) 0.5) t_1))))))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(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[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$1), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[(1.0 + N[(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 17: 43.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)\\ t_1 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_2 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_3 := t_2 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\\ t_4 := \sqrt{t_1 + t_2 \cdot t_3}\\ t_5 := \sqrt{1 - {t_0}^{2}}\\ \mathbf{if}\;t_2 \leq -0.01:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{t_5}\right)\\ \mathbf{elif}\;t_2 \leq 0.0202:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_4}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_1 + t_3 \cdot \left|t_0\right|}}{t_5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (* (- lambda1 lambda2) 0.5)))
        (t_1 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_2 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_3 (* t_2 (* (cos phi1) (cos phi2))))
        (t_4 (sqrt (+ t_1 (* t_2 t_3))))
        (t_5 (sqrt (- 1.0 (pow t_0 2.0)))))
   (if (<= t_2 -0.01)
     (* R (* 2.0 (atan2 t_4 t_5)))
     (if (<= t_2 0.0202)
       (*
        R
        (*
         2.0
         (atan2 t_4 (sqrt (- 1.0 (pow (sin (* 0.5 (- phi1 phi2))) 2.0))))))
       (* R (* 2.0 (atan2 (sqrt (+ t_1 (* t_3 (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 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_2 * (cos(phi1) * cos(phi2));
	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_2 <= -0.01) {
		tmp = R * (2.0 * atan2(t_4, t_5));
	} else if (t_2 <= 0.0202) {
		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_1 + (t_3 * 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(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_2 = sin(((lambda1 - lambda2) / 2.0d0))
    t_3 = t_2 * (cos(phi1) * cos(phi2))
    t_4 = sqrt((t_1 + (t_2 * t_3)))
    t_5 = sqrt((1.0d0 - (t_0 ** 2.0d0)))
    if (t_2 <= (-0.01d0)) then
        tmp = r * (2.0d0 * atan2(t_4, t_5))
    else if (t_2 <= 0.0202d0) 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_1 + (t_3 * 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.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_2 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_3 = t_2 * (Math.cos(phi1) * Math.cos(phi2));
	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_2 <= -0.01) {
		tmp = R * (2.0 * Math.atan2(t_4, t_5));
	} else if (t_2 <= 0.0202) {
		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_1 + (t_3 * 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.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_2 = math.sin(((lambda1 - lambda2) / 2.0))
	t_3 = t_2 * (math.cos(phi1) * math.cos(phi2))
	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_2 <= -0.01:
		tmp = R * (2.0 * math.atan2(t_4, t_5))
	elif t_2 <= 0.0202:
		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_1 + (t_3 * 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(phi1 - phi2) / 2.0)) ^ 2.0
	t_2 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_3 = Float64(t_2 * Float64(cos(phi1) * cos(phi2)))
	t_4 = sqrt(Float64(t_1 + Float64(t_2 * t_3)))
	t_5 = sqrt(Float64(1.0 - (t_0 ^ 2.0)))
	tmp = 0.0
	if (t_2 <= -0.01)
		tmp = Float64(R * Float64(2.0 * atan(t_4, t_5)));
	elseif (t_2 <= 0.0202)
		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_1 + Float64(t_3 * 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(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_2 = sin(((lambda1 - lambda2) / 2.0));
	t_3 = t_2 * (cos(phi1) * cos(phi2));
	t_4 = sqrt((t_1 + (t_2 * t_3)));
	t_5 = sqrt((1.0 - (t_0 ^ 2.0)));
	tmp = 0.0;
	if (t_2 <= -0.01)
		tmp = R * (2.0 * atan2(t_4, t_5));
	elseif (t_2 <= 0.0202)
		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_1 + (t_3 * 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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$2 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Sqrt[N[(t$95$1 + N[(t$95$2 * t$95$3), $MachinePrecision]), $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$2, -0.01], N[(R * N[(2.0 * N[ArcTan[t$95$4 / t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.0202], 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$1 + N[(t$95$3 * N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 18: 62.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\phi_1 \cdot 0.5\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := t_1 \cdot \left(t_1 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)\\ t_3 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_2}\\ t_4 := {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\\ t_5 := \sqrt{1 - \left(t_0 + \cos \phi_1 \cdot t_4\right)}\\ \mathbf{if}\;\phi_1 \leq -6.5 \cdot 10^{-5}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_2}}{t_5}\right)\\ \mathbf{elif}\;\phi_1 \leq 200:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{\sqrt{1 - \left(\cos \phi_2 \cdot t_4 + {\sin \left(\phi_2 \cdot -0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_3}{t_5}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (* phi1 0.5)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* t_1 (* t_1 (* (cos phi1) (cos phi2)))))
        (t_3 (sqrt (+ (pow (sin (/ (- phi1 phi2) 2.0)) 2.0) t_2)))
        (t_4 (pow (sin (* (- lambda1 lambda2) 0.5)) 2.0))
        (t_5 (sqrt (- 1.0 (+ t_0 (* (cos phi1) t_4))))))
   (if (<= phi1 -6.5e-5)
     (* R (* 2.0 (atan2 (sqrt (+ t_0 t_2)) t_5)))
     (if (<= phi1 200.0)
       (*
        R
        (*
         2.0
         (atan2
          t_3
          (sqrt
           (- 1.0 (+ (* (cos phi2) t_4) (pow (sin (* phi2 -0.5)) 2.0)))))))
       (* R (* 2.0 (atan2 t_3 t_5)))))))

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 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = t_1 * (t_1 * (cos(phi1) * cos(phi2)));
	double t_3 = sqrt((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + t_2));
	double t_4 = pow(sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_5 = sqrt((1.0 - (t_0 + (cos(phi1) * t_4))));
	double tmp;
	if (phi1 <= -6.5e-5) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + t_2)), t_5));
	} else if (phi1 <= 200.0) {
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - ((cos(phi2) * t_4) + pow(sin((phi2 * -0.5)), 2.0))))));
	} else {
		tmp = R * (2.0 * atan2(t_3, 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((phi1 * 0.5d0)) ** 2.0d0
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = t_1 * (t_1 * (cos(phi1) * cos(phi2)))
    t_3 = sqrt(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + t_2))
    t_4 = sin(((lambda1 - lambda2) * 0.5d0)) ** 2.0d0
    t_5 = sqrt((1.0d0 - (t_0 + (cos(phi1) * t_4))))
    if (phi1 <= (-6.5d-5)) then
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + t_2)), t_5))
    else if (phi1 <= 200.0d0) then
        tmp = r * (2.0d0 * atan2(t_3, sqrt((1.0d0 - ((cos(phi2) * t_4) + (sin((phi2 * (-0.5d0))) ** 2.0d0))))))
    else
        tmp = r * (2.0d0 * atan2(t_3, 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.pow(Math.sin((phi1 * 0.5)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = t_1 * (t_1 * (Math.cos(phi1) * Math.cos(phi2)));
	double t_3 = Math.sqrt((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_2));
	double t_4 = Math.pow(Math.sin(((lambda1 - lambda2) * 0.5)), 2.0);
	double t_5 = Math.sqrt((1.0 - (t_0 + (Math.cos(phi1) * t_4))));
	double tmp;
	if (phi1 <= -6.5e-5) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + t_2)), t_5));
	} else if (phi1 <= 200.0) {
		tmp = R * (2.0 * Math.atan2(t_3, Math.sqrt((1.0 - ((Math.cos(phi2) * t_4) + Math.pow(Math.sin((phi2 * -0.5)), 2.0))))));
	} else {
		tmp = R * (2.0 * Math.atan2(t_3, t_5));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin((phi1 * 0.5)), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = t_1 * (t_1 * (math.cos(phi1) * math.cos(phi2)))
	t_3 = math.sqrt((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + t_2))
	t_4 = math.pow(math.sin(((lambda1 - lambda2) * 0.5)), 2.0)
	t_5 = math.sqrt((1.0 - (t_0 + (math.cos(phi1) * t_4))))
	tmp = 0
	if phi1 <= -6.5e-5:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + t_2)), t_5))
	elif phi1 <= 200.0:
		tmp = R * (2.0 * math.atan2(t_3, math.sqrt((1.0 - ((math.cos(phi2) * t_4) + math.pow(math.sin((phi2 * -0.5)), 2.0))))))
	else:
		tmp = R * (2.0 * math.atan2(t_3, t_5))
	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) / 2.0))
	t_2 = Float64(t_1 * Float64(t_1 * Float64(cos(phi1) * cos(phi2))))
	t_3 = sqrt(Float64((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + t_2))
	t_4 = sin(Float64(Float64(lambda1 - lambda2) * 0.5)) ^ 2.0
	t_5 = sqrt(Float64(1.0 - Float64(t_0 + Float64(cos(phi1) * t_4))))
	tmp = 0.0
	if (phi1 <= -6.5e-5)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + t_2)), t_5)));
	elseif (phi1 <= 200.0)
		tmp = Float64(R * Float64(2.0 * atan(t_3, sqrt(Float64(1.0 - Float64(Float64(cos(phi2) * t_4) + (sin(Float64(phi2 * -0.5)) ^ 2.0)))))));
	else
		tmp = Float64(R * Float64(2.0 * atan(t_3, t_5)));
	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) / 2.0));
	t_2 = t_1 * (t_1 * (cos(phi1) * cos(phi2)));
	t_3 = sqrt(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + t_2));
	t_4 = sin(((lambda1 - lambda2) * 0.5)) ^ 2.0;
	t_5 = sqrt((1.0 - (t_0 + (cos(phi1) * t_4))));
	tmp = 0.0;
	if (phi1 <= -6.5e-5)
		tmp = R * (2.0 * atan2(sqrt((t_0 + t_2)), t_5));
	elseif (phi1 <= 200.0)
		tmp = R * (2.0 * atan2(t_3, sqrt((1.0 - ((cos(phi2) * t_4) + (sin((phi2 * -0.5)) ^ 2.0))))));
	else
		tmp = R * (2.0 * atan2(t_3, t_5));
	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[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(t$95$1 * N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Sqrt[N[(N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + t$95$2), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$5 = N[Sqrt[N[(1.0 - N[(t$95$0 + N[(N[Cos[phi1], $MachinePrecision] * t$95$4), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -6.5e-5], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 200.0], N[(R * N[(2.0 * N[ArcTan[t$95$3 / N[Sqrt[N[(1.0 - N[(N[(N[Cos[phi2], $MachinePrecision] * t$95$4), $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[t$95$3 / t$95$5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

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

Alternative 19: 50.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\\ \mathbf{if}\;\phi_2 \leq 3.95:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - \left({\sin \left(\phi_1 \cdot 0.5\right)}^{2} + \cos \phi_1 \cdot {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* t_0 (* t_0 (* (cos phi1) (cos phi2))))))))
   (if (<= phi2 3.95)
     (*
      R
      (*
       2.0
       (atan2
        t_1
        (sqrt
         (-
          1.0
          (+
           (pow (sin (* phi1 0.5)) 2.0)
           (* (cos phi1) (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((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))));
	double tmp;
	if (phi2 <= 3.95) {
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - (pow(sin((phi1 * 0.5)), 2.0) + (cos(phi1) * 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(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))))
    if (phi2 <= 3.95d0) then
        tmp = r * (2.0d0 * atan2(t_1, sqrt((1.0d0 - ((sin((phi1 * 0.5d0)) ** 2.0d0) + (cos(phi1) * (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((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2))))));
	double tmp;
	if (phi2 <= 3.95) {
		tmp = R * (2.0 * Math.atan2(t_1, Math.sqrt((1.0 - (Math.pow(Math.sin((phi1 * 0.5)), 2.0) + (Math.cos(phi1) * 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((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2))))))
	tmp = 0
	if phi2 <= 3.95:
		tmp = R * (2.0 * math.atan2(t_1, math.sqrt((1.0 - (math.pow(math.sin((phi1 * 0.5)), 2.0) + (math.cos(phi1) * 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((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2))))))
	tmp = 0.0
	if (phi2 <= 3.95)
		tmp = Float64(R * Float64(2.0 * atan(t_1, sqrt(Float64(1.0 - Float64((sin(Float64(phi1 * 0.5)) ^ 2.0) + Float64(cos(phi1) * (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(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))));
	tmp = 0.0;
	if (phi2 <= 3.95)
		tmp = R * (2.0 * atan2(t_1, sqrt((1.0 - ((sin((phi1 * 0.5)) ^ 2.0) + (cos(phi1) * (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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $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]], $MachinePrecision]}, If[LessEqual[phi2, 3.95], N[(R * N[(2.0 * N[ArcTan[t$95$1 / N[Sqrt[N[(1.0 - N[(N[Power[N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] + N[(N[Cos[phi1], $MachinePrecision] * N[Power[N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $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 20: 43.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_1 := \sqrt{{\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2} + t_0 \cdot \left(t_0 \cdot \left(\cos \phi_1 \cdot \cos \phi_2\right)\right)}\\ \mathbf{if}\;t_0 \leq -0.01 \lor \neg \left(t_0 \leq 0.0202\right):\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(\left(\lambda_1 - \lambda_2\right) \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{t_1}{\sqrt{1 - {\sin \left(0.5 \cdot \left(\phi_1 - \phi_2\right)\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_1
         (sqrt
          (+
           (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
           (* t_0 (* t_0 (* (cos phi1) (cos phi2))))))))
   (if (or (<= t_0 -0.01) (not (<= t_0 0.0202)))
     (*
      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((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))));
	double tmp;
	if ((t_0 <= -0.01) || !(t_0 <= 0.0202)) {
		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(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))))
    if ((t_0 <= (-0.01d0)) .or. (.not. (t_0 <= 0.0202d0))) 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((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2))))));
	double tmp;
	if ((t_0 <= -0.01) || !(t_0 <= 0.0202)) {
		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((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2))))))
	tmp = 0
	if (t_0 <= -0.01) or not (t_0 <= 0.0202):
		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((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2))))))
	tmp = 0.0
	if ((t_0 <= -0.01) || !(t_0 <= 0.0202))
		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(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2))))));
	tmp = 0.0;
	if ((t_0 <= -0.01) || ~((t_0 <= 0.0202)))
		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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $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]], $MachinePrecision]}, If[Or[LessEqual[t$95$0, -0.01], N[Not[LessEqual[t$95$0, 0.0202]], $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 21: 34.4% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* t_0 (* t_0 (* (cos phi1) (cos phi2))))))
      (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((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2)))))), 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(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2)))))), 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((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)))))), 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((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))))), 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((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2)))))), 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(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2)))))), 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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $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]], $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 22: 32.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\sin \left(\frac{\phi_1 - \phi_2}{2}\right)}^{2}\\ t_1 := \sin \left(\frac{\lambda_1 - \lambda_2}{2}\right)\\ t_2 := \cos \phi_1 \cdot \cos \phi_2\\ \mathbf{if}\;\lambda_1 \leq 0.0029:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + t_1 \cdot \left(t_1 \cdot t_2\right)}}{\sqrt{{\cos \left(\lambda_2 \cdot 0.5\right)}^{2}}}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(2 \cdot \tan^{-1}_* \frac{\sqrt{t_0 + {\sin \left(\lambda_1 \cdot 0.5\right)}^{2} \cdot t_2}}{\sqrt{{\cos \left(\lambda_1 \cdot 0.5\right)}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (pow (sin (/ (- phi1 phi2) 2.0)) 2.0))
        (t_1 (sin (/ (- lambda1 lambda2) 2.0)))
        (t_2 (* (cos phi1) (cos phi2))))
   (if (<= lambda1 0.0029)
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* t_1 (* t_1 t_2))))
        (sqrt (pow (cos (* lambda2 0.5)) 2.0)))))
     (*
      R
      (*
       2.0
       (atan2
        (sqrt (+ t_0 (* (pow (sin (* lambda1 0.5)) 2.0) t_2)))
        (sqrt (pow (cos (* lambda1 0.5)) 2.0))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = pow(sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = sin(((lambda1 - lambda2) / 2.0));
	double t_2 = cos(phi1) * cos(phi2);
	double tmp;
	if (lambda1 <= 0.0029) {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_1 * (t_1 * t_2)))), sqrt(pow(cos((lambda2 * 0.5)), 2.0))));
	} else {
		tmp = R * (2.0 * atan2(sqrt((t_0 + (pow(sin((lambda1 * 0.5)), 2.0) * t_2))), sqrt(pow(cos((lambda1 * 0.5)), 2.0))));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0
    t_1 = sin(((lambda1 - lambda2) / 2.0d0))
    t_2 = cos(phi1) * cos(phi2)
    if (lambda1 <= 0.0029d0) then
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + (t_1 * (t_1 * t_2)))), sqrt((cos((lambda2 * 0.5d0)) ** 2.0d0))))
    else
        tmp = r * (2.0d0 * atan2(sqrt((t_0 + ((sin((lambda1 * 0.5d0)) ** 2.0d0) * t_2))), sqrt((cos((lambda1 * 0.5d0)) ** 2.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0);
	double t_1 = Math.sin(((lambda1 - lambda2) / 2.0));
	double t_2 = Math.cos(phi1) * Math.cos(phi2);
	double tmp;
	if (lambda1 <= 0.0029) {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (t_1 * (t_1 * t_2)))), Math.sqrt(Math.pow(Math.cos((lambda2 * 0.5)), 2.0))));
	} else {
		tmp = R * (2.0 * Math.atan2(Math.sqrt((t_0 + (Math.pow(Math.sin((lambda1 * 0.5)), 2.0) * t_2))), Math.sqrt(Math.pow(Math.cos((lambda1 * 0.5)), 2.0))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0)
	t_1 = math.sin(((lambda1 - lambda2) / 2.0))
	t_2 = math.cos(phi1) * math.cos(phi2)
	tmp = 0
	if lambda1 <= 0.0029:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (t_1 * (t_1 * t_2)))), math.sqrt(math.pow(math.cos((lambda2 * 0.5)), 2.0))))
	else:
		tmp = R * (2.0 * math.atan2(math.sqrt((t_0 + (math.pow(math.sin((lambda1 * 0.5)), 2.0) * t_2))), math.sqrt(math.pow(math.cos((lambda1 * 0.5)), 2.0))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0
	t_1 = sin(Float64(Float64(lambda1 - lambda2) / 2.0))
	t_2 = Float64(cos(phi1) * cos(phi2))
	tmp = 0.0
	if (lambda1 <= 0.0029)
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64(t_1 * Float64(t_1 * t_2)))), sqrt((cos(Float64(lambda2 * 0.5)) ^ 2.0)))));
	else
		tmp = Float64(R * Float64(2.0 * atan(sqrt(Float64(t_0 + Float64((sin(Float64(lambda1 * 0.5)) ^ 2.0) * t_2))), sqrt((cos(Float64(lambda1 * 0.5)) ^ 2.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin(((phi1 - phi2) / 2.0)) ^ 2.0;
	t_1 = sin(((lambda1 - lambda2) / 2.0));
	t_2 = cos(phi1) * cos(phi2);
	tmp = 0.0;
	if (lambda1 <= 0.0029)
		tmp = R * (2.0 * atan2(sqrt((t_0 + (t_1 * (t_1 * t_2)))), sqrt((cos((lambda2 * 0.5)) ^ 2.0))));
	else
		tmp = R * (2.0 * atan2(sqrt((t_0 + ((sin((lambda1 * 0.5)) ^ 2.0) * t_2))), sqrt((cos((lambda1 * 0.5)) ^ 2.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[Sin[N[(N[(lambda1 - lambda2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Cos[phi1], $MachinePrecision] * N[Cos[phi2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, 0.0029], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(t$95$1 * N[(t$95$1 * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Power[N[Cos[N[(lambda2 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(2.0 * N[ArcTan[N[Sqrt[N[(t$95$0 + N[(N[Power[N[Sin[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[N[Power[N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

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

Alternative 23: 29.7% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (sin (/ (- lambda1 lambda2) 2.0))))
   (*
    R
    (*
     2.0
     (atan2
      (sqrt
       (+
        (pow (sin (/ (- phi1 phi2) 2.0)) 2.0)
        (* t_0 (* t_0 (* (cos phi1) (cos phi2))))))
      (sqrt (pow (cos (* lambda1 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((pow(sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2)))))), sqrt(pow(cos((lambda1 * 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(((sin(((phi1 - phi2) / 2.0d0)) ** 2.0d0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2)))))), sqrt((cos((lambda1 * 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((Math.pow(Math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (Math.cos(phi1) * Math.cos(phi2)))))), Math.sqrt(Math.pow(Math.cos((lambda1 * 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((math.pow(math.sin(((phi1 - phi2) / 2.0)), 2.0) + (t_0 * (t_0 * (math.cos(phi1) * math.cos(phi2)))))), math.sqrt(math.pow(math.cos((lambda1 * 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((sin(Float64(Float64(phi1 - phi2) / 2.0)) ^ 2.0) + Float64(t_0 * Float64(t_0 * Float64(cos(phi1) * cos(phi2)))))), sqrt((cos(Float64(lambda1 * 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(((sin(((phi1 - phi2) / 2.0)) ^ 2.0) + (t_0 * (t_0 * (cos(phi1) * cos(phi2)))))), sqrt((cos((lambda1 * 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[Power[N[Sin[N[(N[(phi1 - phi2), $MachinePrecision] / 2.0), $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]], $MachinePrecision] / N[Sqrt[N[Power[N[Cos[N[(lambda1 * 0.5), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

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

Alternative 24: 29.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Alternative 25: 29.7% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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

Reproduce

herbie shell --seed 2024008 
(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))))))))))