Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.0% → 99.9%

Time: 19.3s

Alternatives: 13

Speedup: 3.0×

• 30.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0d0))
    code = r * sqrt(((t_0 * t_0) + ((phi1 - phi2) * (phi1 - phi2))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(0.5 \cdot \phi_2\right)\right)\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot
   (*
    (- lambda1 lambda2)
    (fma
     (cos (* 0.5 phi2))
     (cos (* 0.5 phi1))
     (* (sin (* 0.5 phi1)) (- (sin (* 0.5 phi2))))))
   (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * fma(cos((0.5 * phi2)), cos((0.5 * phi1)), (sin((0.5 * phi1)) * -sin((0.5 * phi2))))), (phi1 - phi2));
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * fma(cos(Float64(0.5 * phi2)), cos(Float64(0.5 * phi1)), Float64(sin(Float64(0.5 * phi1)) * Float64(-sin(Float64(0.5 * phi2)))))), Float64(phi1 - phi2)))
end

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. hypot-def 95.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Step-by-step derivation

   1. add-log-exp 95.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

   2. div-inv 95.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

   3. metadata-eval 95.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}\right), \phi_1 - \phi_2\right) \]

5. Applied egg-rr 95.7%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 95.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}}\right), \phi_1 - \phi_2\right) \]

   2. +-commutative 95.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

   3. distribute-lft-in 95.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \phi_2 + 0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

   4. cos-sum 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

   5. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

   6. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

   7. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

   8. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]

7. Applied egg-rr 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
8. Step-by-step derivation

   1. fma-neg 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}}\right), \phi_1 - \phi_2\right) \]

   2. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

   3. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}, -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

   4. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

   5. *-commutative 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \color{blue}{\left(0.5 \cdot \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

9. Simplified 99.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}\right), \phi_1 - \phi_2\right) \]
10. Step-by-step derivation

    1. add-cbrt-cube 99.8%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\color{blue}{\sqrt[3]{\left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)}} \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}\right), \phi_1 - \phi_2\right) \]

    2. add-cbrt-cube 99.8%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sqrt[3]{\left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)} \cdot \color{blue}{\sqrt[3]{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}}\right)}\right), \phi_1 - \phi_2\right) \]

    3. cbrt-unprod 99.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\color{blue}{\sqrt[3]{\left(\left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \left(\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}\right)}\right), \phi_1 - \phi_2\right) \]

    4. pow3 99.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sqrt[3]{\color{blue}{{\sin \left(0.5 \cdot \phi_2\right)}^{3}} \cdot \left(\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}\right)}\right), \phi_1 - \phi_2\right) \]

    5. pow3 99.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sqrt[3]{{\sin \left(0.5 \cdot \phi_2\right)}^{3} \cdot \color{blue}{{\sin \left(0.5 \cdot \phi_1\right)}^{3}}}\right)}\right), \phi_1 - \phi_2\right) \]

11. Applied egg-rr 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\color{blue}{\sqrt[3]{{\sin \left(0.5 \cdot \phi_2\right)}^{3} \cdot {\sin \left(0.5 \cdot \phi_1\right)}^{3}}}\right)}\right), \phi_1 - \phi_2\right) \]
12. Step-by-step derivation

    1. add-log-exp 99.9%

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

    2. cbrt-prod 99.8%

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

    3. unpow3 99.9%

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

    4. add-cbrt-cube 99.9%

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

    5. unpow3 99.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sqrt[3]{\color{blue}{\left(\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

    6. add-cbrt-cube 99.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \color{blue}{\sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

13. Applied egg-rr 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}, \phi_1 - \phi_2\right) \]

14. Final simplification 99.9%

    \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_1\right) \cdot \left(-\sin \left(0.5 \cdot \phi_2\right)\right)\right), \phi_1 - \phi_2\right) \]

Alternative 2: 96.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{+185}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.7e+185)
   (*
    R
    (hypot
     (*
      lambda1
      (-
       (* (cos (* 0.5 phi2)) (cos (* 0.5 phi1)))
       (* (sin (* 0.5 phi2)) (sin (* 0.5 phi1)))))
     (- phi1 phi2)))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (log (exp (cos (* 0.5 (+ phi2 phi1))))))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.7e+185) {
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi2)) * cos((0.5 * phi1))) - (sin((0.5 * phi2)) * sin((0.5 * phi1))))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * log(exp(cos((0.5 * (phi2 + phi1)))))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.7e+185) {
		tmp = R * Math.hypot((lambda1 * ((Math.cos((0.5 * phi2)) * Math.cos((0.5 * phi1))) - (Math.sin((0.5 * phi2)) * Math.sin((0.5 * phi1))))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.log(Math.exp(Math.cos((0.5 * (phi2 + phi1)))))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.7e+185:
		tmp = R * math.hypot((lambda1 * ((math.cos((0.5 * phi2)) * math.cos((0.5 * phi1))) - (math.sin((0.5 * phi2)) * math.sin((0.5 * phi1))))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.log(math.exp(math.cos((0.5 * (phi2 + phi1)))))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.7e+185)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(Float64(cos(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * log(exp(cos(Float64(0.5 * Float64(phi2 + phi1)))))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.7e+185)
		tmp = R * hypot((lambda1 * ((cos((0.5 * phi2)) * cos((0.5 * phi1))) - (sin((0.5 * phi2)) * sin((0.5 * phi1))))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * log(exp(cos((0.5 * (phi2 + phi1)))))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.7e+185], N[(R * N[Sqrt[N[(lambda1 * N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Log[N[Exp[N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.70000000000000009e185

   1. Initial program 48.2%

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

      1. hypot-def 88.3%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 88.3%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 88.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 88.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 88.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}\right), \phi_1 - \phi_2\right) \]

   5. Applied egg-rr 88.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 88.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}}\right), \phi_1 - \phi_2\right) \]

      2. +-commutative 88.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

      3. distribute-lft-in 88.2%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \phi_2 + 0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

      4. cos-sum 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

      5. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      6. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      7. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      8. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 99.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. fma-neg 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}}\right), \phi_1 - \phi_2\right) \]

      2. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      3. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}, -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      4. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      5. *-commutative 99.7%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \color{blue}{\left(0.5 \cdot \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

   9. Simplified 99.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}\right), \phi_1 - \phi_2\right) \]

   10. Taylor expanded in lambda1 around inf 95.5%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1}, \phi_1 - \phi_2\right) \]

   if -1.70000000000000009e185 < lambda1

   1. Initial program 56.6%

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

      1. hypot-def 96.8%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.8%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 96.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 96.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 96.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}\right), \phi_1 - \phi_2\right) \]

   5. Applied egg-rr 96.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.7 \cdot 10^{+185}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right)}\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 3: 86.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\\ t_1 := \cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(t_1 - t_0\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(t_0 - t_1\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (sin (* 0.5 phi2)) (sin (* 0.5 phi1))))
        (t_1 (* (cos (* 0.5 phi2)) (cos (* 0.5 phi1)))))
   (if (<= lambda1 -1.65e+51)
     (* R (hypot (* lambda1 (- t_1 t_0)) (- phi1 phi2)))
     (* R (hypot (* lambda2 (- t_0 t_1)) (- phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = sin((0.5 * phi2)) * sin((0.5 * phi1));
	double t_1 = cos((0.5 * phi2)) * cos((0.5 * phi1));
	double tmp;
	if (lambda1 <= -1.65e+51) {
		tmp = R * hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.sin((0.5 * phi2)) * Math.sin((0.5 * phi1));
	double t_1 = Math.cos((0.5 * phi2)) * Math.cos((0.5 * phi1));
	double tmp;
	if (lambda1 <= -1.65e+51) {
		tmp = R * Math.hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.sin((0.5 * phi2)) * math.sin((0.5 * phi1))
	t_1 = math.cos((0.5 * phi2)) * math.cos((0.5 * phi1))
	tmp = 0
	if lambda1 <= -1.65e+51:
		tmp = R * math.hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(sin(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1)))
	t_1 = Float64(cos(Float64(0.5 * phi2)) * cos(Float64(0.5 * phi1)))
	tmp = 0.0
	if (lambda1 <= -1.65e+51)
		tmp = Float64(R * hypot(Float64(lambda1 * Float64(t_1 - t_0)), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(t_0 - t_1)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = sin((0.5 * phi2)) * sin((0.5 * phi1));
	t_1 = cos((0.5 * phi2)) * cos((0.5 * phi1));
	tmp = 0.0;
	if (lambda1 <= -1.65e+51)
		tmp = R * hypot((lambda1 * (t_1 - t_0)), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * (t_0 - t_1)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[lambda1, -1.65e+51], N[(R * N[Sqrt[N[(lambda1 * N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.6499999999999999e51

   1. Initial program 45.7%

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

      1. hypot-def 93.6%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.6%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 93.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 93.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 93.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}\right), \phi_1 - \phi_2\right) \]

   5. Applied egg-rr 93.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 93.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}}\right), \phi_1 - \phi_2\right) \]

      2. +-commutative 93.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

      3. distribute-lft-in 93.6%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \phi_2 + 0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

      4. cos-sum 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

      5. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      6. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      7. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      8. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. fma-neg 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}}\right), \phi_1 - \phi_2\right) \]

      2. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      3. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}, -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      4. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      5. *-commutative 99.8%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \color{blue}{\left(0.5 \cdot \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

   9. Simplified 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}\right), \phi_1 - \phi_2\right) \]

   10. Taylor expanded in lambda1 around inf 92.8%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_1}, \phi_1 - \phi_2\right) \]

   if -1.6499999999999999e51 < lambda1

   1. Initial program 58.8%

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

      1. hypot-def 96.4%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.4%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
   4. Step-by-step derivation

      1. add-log-exp 96.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right)}, \phi_1 - \phi_2\right) \]

      2. div-inv 96.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

      3. metadata-eval 96.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right)}\right), \phi_1 - \phi_2\right) \]

   5. Applied egg-rr 96.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\log \left(e^{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 96.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}}\right), \phi_1 - \phi_2\right) \]

      2. +-commutative 96.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(0.5 \cdot \color{blue}{\left(\phi_2 + \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

      3. distribute-lft-in 96.4%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(0.5 \cdot \phi_2 + 0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

      4. cos-sum 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}}\right), \phi_1 - \phi_2\right) \]

      5. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      6. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      7. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_1\right)}\right), \phi_1 - \phi_2\right) \]

      8. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]

   7. Applied egg-rr 99.9%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)}}\right), \phi_1 - \phi_2\right) \]
   8. Step-by-step derivation

      1. fma-neg 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right), \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}}\right), \phi_1 - \phi_2\right) \]

      2. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \color{blue}{\left(0.5 \cdot \phi_2\right)}, \cos \left(\phi_1 \cdot 0.5\right), -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      3. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \color{blue}{\left(0.5 \cdot \phi_1\right)}, -\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      4. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \color{blue}{\left(0.5 \cdot \phi_2\right)} \cdot \sin \left(\phi_1 \cdot 0.5\right)\right)}\right), \phi_1 - \phi_2\right) \]

      5. *-commutative 99.9%

         \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \color{blue}{\left(0.5 \cdot \phi_1\right)}\right)}\right), \phi_1 - \phi_2\right) \]

   9. Simplified 99.9%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \log \left(e^{\color{blue}{\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), -\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}}\right), \phi_1 - \phi_2\right) \]

   10. Taylor expanded in lambda1 around 0 88.3%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-1 \cdot \left(\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right)}, \phi_1 - \phi_2\right) \]
   11. Step-by-step derivation

       1. mul-1-neg 88.3%

          \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-\lambda_2 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)}, \phi_1 - \phi_2\right) \]

       2. *-commutative 88.3%

          \[\leadsto R \cdot \mathsf{hypot}\left(-\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2}, \phi_1 - \phi_2\right) \]

       3. distribute-rgt-neg-in 88.3%

          \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(-\lambda_2\right)}, \phi_1 - \phi_2\right) \]

       4. *-commutative 88.3%

          \[\leadsto R \cdot \mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \color{blue}{\sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)}\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right) \]

   12. Simplified 88.3%

       \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right) \cdot \left(-\lambda_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 4: 92.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.05e-7)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.05e-7) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.05e-7) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.05e-7:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.05e-7)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.05e-7)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.05e-7], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.05e-7

   1. Initial program 40.2%

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

      1. hypot-def 88.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 88.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around 0 88.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   if -1.05e-7 < phi1

   1. Initial program 60.3%

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

      1. hypot-def 98.1%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 93.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{-7}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 5: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
}

Java

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

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
end

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. hypot-def 95.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

4. Final simplification 95.7%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right) \]

Alternative 6: 78.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.2 \cdot 10^{+89}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -5.2e+89)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda2)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5.2e+89) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot((cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5.2e+89) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -5.2e+89:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * -lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -5.2e+89)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(-lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -5.2e+89)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot((cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -5.2e+89], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * (-lambda2)), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -5.2000000000000001e89

   1. Initial program 45.3%

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

      1. hypot-def 92.4%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 92.4%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 83.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   5. Taylor expanded in phi2 around 0 45.4%

      \[\leadsto \color{blue}{\sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \cdot R} \]
   6. Step-by-step derivation

      1. *-commutative 45.4%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      2. unpow2 45.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. unpow2 45.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      4. hypot-def 70.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)} \]

   7. Simplified 70.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)} \]

   if -5.2000000000000001e89 < lambda1

   1. Initial program 58.2%

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

      1. hypot-def 96.5%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around 0 91.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

   5. Taylor expanded in lambda1 around 0 82.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-1 \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. mul-1-neg 82.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{-\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

      2. distribute-lft-neg-out 82.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\left(-\lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

      3. *-commutative 82.5%

         \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)}, \phi_1 - \phi_2\right) \]

   7. Simplified 82.5%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right)}, \phi_1 - \phi_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5.2 \cdot 10^{+89}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]

Alternative 7: 90.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right) \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)))
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
end

Wolfram

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

Derivation

1. Initial program 55.5%

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

   1. hypot-def 95.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

4. Taylor expanded in phi2 around 0 90.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_1\right)}, \phi_1 - \phi_2\right) \]

5. Final simplification 90.8%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right) \]

Alternative 8: 70.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.45 \cdot 10^{+52}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.45e+52)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.45e+52) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.45e+52) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.45e+52:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.45e+52)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.45e+52)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.45e+52], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.45e52

   1. Initial program 57.6%

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

      1. hypot-def 96.5%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 90.4%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   5. Taylor expanded in phi2 around 0 50.4%

      \[\leadsto \color{blue}{\sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \cdot R} \]
   6. Step-by-step derivation

      1. *-commutative 50.4%

         \[\leadsto \color{blue}{R \cdot \sqrt{{\phi_1}^{2} + {\left(\lambda_1 - \lambda_2\right)}^{2}}} \]

      2. unpow2 50.4%

         \[\leadsto R \cdot \sqrt{\color{blue}{\phi_1 \cdot \phi_1} + {\left(\lambda_1 - \lambda_2\right)}^{2}} \]

      3. unpow2 50.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}} \]

      4. hypot-def 71.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)} \]

   7. Simplified 71.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)} \]

   if 1.45e52 < phi2

   1. Initial program 48.0%

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

      1. hypot-def 92.9%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 92.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 62.3%

      \[\leadsto \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. *-commutative 62.3%

         \[\leadsto \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]

      2. associate-*r* 62.3%

         \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. distribute-rgt-out 67.5%

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)} \]

      4. mul-1-neg 67.5%

         \[\leadsto R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

      5. unsub-neg 67.5%

         \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   6. Simplified 67.5%

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.45 \cdot 10^{+52}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 9: 27.5% accurate, 26.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-74} \lor \neg \left(\phi_1 \leq -2.06 \cdot 10^{-165}\right) \land \phi_1 \leq -1.05 \cdot 10^{-279}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.5e+19)
   (* R (- phi1))
   (if (or (<= phi1 -2.4e-74)
           (and (not (<= phi1 -2.06e-165)) (<= phi1 -1.05e-279)))
     (* lambda1 (- R))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e+19) {
		tmp = R * -phi1;
	} else if ((phi1 <= -2.4e-74) || (!(phi1 <= -2.06e-165) && (phi1 <= -1.05e-279))) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * phi2;
	}
	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) :: tmp
    if (phi1 <= (-5.5d+19)) then
        tmp = r * -phi1
    else if ((phi1 <= (-2.4d-74)) .or. (.not. (phi1 <= (-2.06d-165))) .and. (phi1 <= (-1.05d-279))) then
        tmp = lambda1 * -r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e+19) {
		tmp = R * -phi1;
	} else if ((phi1 <= -2.4e-74) || (!(phi1 <= -2.06e-165) && (phi1 <= -1.05e-279))) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.5e+19:
		tmp = R * -phi1
	elif (phi1 <= -2.4e-74) or (not (phi1 <= -2.06e-165) and (phi1 <= -1.05e-279)):
		tmp = lambda1 * -R
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.5e+19)
		tmp = Float64(R * Float64(-phi1));
	elseif ((phi1 <= -2.4e-74) || (!(phi1 <= -2.06e-165) && (phi1 <= -1.05e-279)))
		tmp = Float64(lambda1 * Float64(-R));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.5e+19)
		tmp = R * -phi1;
	elseif ((phi1 <= -2.4e-74) || (~((phi1 <= -2.06e-165)) && (phi1 <= -1.05e-279)))
		tmp = lambda1 * -R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.5e+19], N[(R * (-phi1)), $MachinePrecision], If[Or[LessEqual[phi1, -2.4e-74], And[N[Not[LessEqual[phi1, -2.06e-165]], $MachinePrecision], LessEqual[phi1, -1.05e-279]]], N[(lambda1 * (-R)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -5.5e19

   1. Initial program 38.8%

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

      1. hypot-def 88.8%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 88.8%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 68.4%

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 68.4%

         \[\leadsto \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      2. mul-1-neg 68.4%

         \[\leadsto \color{blue}{\left(-\phi_1\right)} \cdot R \]

   6. Simplified 68.4%

      \[\leadsto \color{blue}{\left(-\phi_1\right) \cdot R} \]

   if -5.5e19 < phi1 < -2.3999999999999999e-74 or -2.05999999999999988e-165 < phi1 < -1.05000000000000003e-279

   1. Initial program 64.0%

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

      1. hypot-def 97.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 96.7%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   5. Taylor expanded in lambda1 around -inf 20.9%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 20.9%

         \[\leadsto \color{blue}{-R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]

      2. distribute-rgt-neg-in 20.9%

         \[\leadsto \color{blue}{R \cdot \left(-\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]

      3. distribute-rgt-neg-in 20.9%

         \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)} \]

   7. Simplified 20.9%

      \[\leadsto \color{blue}{R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)} \]

   8. Taylor expanded in phi2 around 0 20.9%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 20.9%

         \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

      2. *-commutative 20.9%

         \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

      3. distribute-rgt-neg-in 20.9%

         \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   10. Simplified 20.9%

       \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   if -2.3999999999999999e-74 < phi1 < -2.05999999999999988e-165 or -1.05000000000000003e-279 < phi1

   1. Initial program 59.6%

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

      1. hypot-def 97.9%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 97.9%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around inf 20.0%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 20.0%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   6. Simplified 20.0%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{+19}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-74} \lor \neg \left(\phi_1 \leq -2.06 \cdot 10^{-165}\right) \land \phi_1 \leq -1.05 \cdot 10^{-279}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 10: 33.9% accurate, 46.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+146}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.65e+146) (* lambda1 (- R)) (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.65e+146) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	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) :: tmp
    if (lambda1 <= (-1.65d+146)) then
        tmp = lambda1 * -r
    else
        tmp = r * (phi2 - phi1)
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.65e+146) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.65e+146:
		tmp = lambda1 * -R
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.65e+146)
		tmp = Float64(lambda1 * Float64(-R));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.65e+146)
		tmp = lambda1 * -R;
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.65e+146], N[(lambda1 * (-R)), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -1.65000000000000008e146

   1. Initial program 45.7%

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

      1. hypot-def 90.7%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 90.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 80.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   5. Taylor expanded in lambda1 around -inf 49.8%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 49.8%

         \[\leadsto \color{blue}{-R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]

      2. distribute-rgt-neg-in 49.8%

         \[\leadsto \color{blue}{R \cdot \left(-\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]

      3. distribute-rgt-neg-in 49.8%

         \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)} \]

   7. Simplified 49.8%

      \[\leadsto \color{blue}{R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)} \]

   8. Taylor expanded in phi2 around 0 62.8%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 62.8%

         \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

      2. *-commutative 62.8%

         \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

      3. distribute-rgt-neg-in 62.8%

         \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   10. Simplified 62.8%

       \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   if -1.65000000000000008e146 < lambda1

   1. Initial program 57.5%

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

      1. hypot-def 96.7%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.7%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around -inf 33.4%

      \[\leadsto \color{blue}{R \cdot \phi_2 + -1 \cdot \left(\phi_1 \cdot R\right)} \]
   5. Step-by-step derivation

      1. *-commutative 33.4%

         \[\leadsto \color{blue}{\phi_2 \cdot R} + -1 \cdot \left(\phi_1 \cdot R\right) \]

      2. associate-*r* 33.4%

         \[\leadsto \phi_2 \cdot R + \color{blue}{\left(-1 \cdot \phi_1\right) \cdot R} \]

      3. distribute-rgt-out 34.4%

         \[\leadsto \color{blue}{R \cdot \left(\phi_2 + -1 \cdot \phi_1\right)} \]

      4. mul-1-neg 34.4%

         \[\leadsto R \cdot \left(\phi_2 + \color{blue}{\left(-\phi_1\right)}\right) \]

      5. unsub-neg 34.4%

         \[\leadsto R \cdot \color{blue}{\left(\phi_2 - \phi_1\right)} \]

   6. Simplified 34.4%

      \[\leadsto \color{blue}{R \cdot \left(\phi_2 - \phi_1\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+146}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 11: 25.5% accurate, 54.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.4e+36) (* lambda1 (- R)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.4e+36) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * phi2;
	}
	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) :: tmp
    if (phi2 <= 3.4d+36) then
        tmp = lambda1 * -r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.4e+36) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.4e+36:
		tmp = lambda1 * -R
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.4e+36)
		tmp = Float64(lambda1 * Float64(-R));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.4e+36)
		tmp = lambda1 * -R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.4e+36], N[(lambda1 * (-R)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.3999999999999998e36

   1. Initial program 57.9%

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

      1. hypot-def 96.5%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 96.5%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi1 around 0 90.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]

   5. Taylor expanded in lambda1 around -inf 17.1%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 17.1%

         \[\leadsto \color{blue}{-R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]

      2. distribute-rgt-neg-in 17.1%

         \[\leadsto \color{blue}{R \cdot \left(-\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]

      3. distribute-rgt-neg-in 17.1%

         \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)} \]

   7. Simplified 17.1%

      \[\leadsto \color{blue}{R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(-\lambda_1\right)\right)} \]

   8. Taylor expanded in phi2 around 0 18.4%

      \[\leadsto \color{blue}{-1 \cdot \left(R \cdot \lambda_1\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 18.4%

         \[\leadsto \color{blue}{-R \cdot \lambda_1} \]

      2. *-commutative 18.4%

         \[\leadsto -\color{blue}{\lambda_1 \cdot R} \]

      3. distribute-rgt-neg-in 18.4%

         \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   10. Simplified 18.4%

       \[\leadsto \color{blue}{\lambda_1 \cdot \left(-R\right)} \]

   if 3.3999999999999998e36 < phi2

   1. Initial program 47.3%

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

      1. hypot-def 93.0%

         \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   3. Simplified 93.0%

      \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

   4. Taylor expanded in phi2 around inf 60.5%

      \[\leadsto \color{blue}{R \cdot \phi_2} \]
   5. Step-by-step derivation

      1. *-commutative 60.5%

         \[\leadsto \color{blue}{\phi_2 \cdot R} \]

   6. Simplified 60.5%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.4 \cdot 10^{+36}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12: 14.1% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \lambda_1 \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R lambda1))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * lambda1;
}

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 * lambda1
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * lambda1;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * lambda1

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * lambda1)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * lambda1;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda1), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

   1. hypot-def 95.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

4. Taylor expanded in phi1 around 0 90.9%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\cos \left(0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right) \]
5. Step-by-step derivation

   1. expm1-log1p-u 57.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right) \]

   2. *-commutative 57.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)\right), \phi_1 - \phi_2\right) \]

6. Applied egg-rr 57.3%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\right)}, \phi_1 - \phi_2\right) \]

7. Taylor expanded in lambda1 around inf 13.7%

   \[\leadsto \color{blue}{R \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right)} \]
8. Step-by-step derivation

   1. *-commutative 13.7%

      \[\leadsto \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_1\right) \cdot R} \]

   2. associate-*l* 13.7%

      \[\leadsto \color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 \cdot R\right)} \]

9. Simplified 13.7%

   \[\leadsto \color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 \cdot R\right)} \]

10. Taylor expanded in phi2 around 0 11.5%

    \[\leadsto \color{blue}{R \cdot \lambda_1} \]

11. Final simplification 11.5%

    \[\leadsto R \cdot \lambda_1 \]

Alternative 13: 17.2% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \phi_2 \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* R phi2))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

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 * phi2
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * phi2;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	return R * phi2

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * phi2)
end

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * phi2;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 55.5%

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

   1. hypot-def 95.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

3. Simplified 95.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]

4. Taylor expanded in phi2 around inf 16.2%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
5. Step-by-step derivation

   1. *-commutative 16.2%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]

6. Simplified 16.2%

   \[\leadsto \color{blue}{\phi_2 \cdot R} \]

7. Final simplification 16.2%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023200 
(FPCore (R lambda1 lambda2 phi1 phi2)
  :name "Equirectangular approximation to distance on a great circle"
  :precision binary64
  (* R (sqrt (+ (* (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0)))) (* (- phi1 phi2) (- phi1 phi2))))))