Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.1% → 99.9%

Time: 32.0s

Alternatives: 14

Speedup: 2.9×

• Unsound rule application detected in e-graph. Results may not be sound.

• 31.5% 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: 60.1% 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, 0.6× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi1 * 0.5), $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 61.5%

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

   1. add-sqr-sqrt 61.2%

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

   2. pow2 61.2%

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

3. Applied egg-rr 96.4%

   \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
4. Step-by-step derivation

   1. *-commutative 96.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   2. distribute-rgt-in 96.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(\phi_1 \cdot 0.5 + \phi_2 \cdot 0.5\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   3. *-commutative 96.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\color{blue}{0.5 \cdot \phi_1} + \phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   4. cos-sum 99.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   5. *-commutative 99.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   6. *-commutative 99.4%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

5. Applied egg-rr 99.4%

   \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]
6. Step-by-step derivation

   1. cancel-sign-sub-inv 99.4%

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

   2. add-cube-cbrt 99.3%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)} \cdot \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}\right) \cdot \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}} + \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   3. fma-def 99.3%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)} \cdot \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]

   4. pow2 99.3%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}\right)}^{2}}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   5. *-commutative 99.3%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_2\right)}}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   6. *-commutative 99.3%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_2\right)}}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   7. *-commutative 99.3%

      \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \color{blue}{\left(0.5 \cdot \phi_2\right)}\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

7. Applied egg-rr 99.3%

   \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)}\right)}^{2} \]
8. Step-by-step derivation

   1. unpow2 99.3%

      \[\leadsto R \cdot \color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)}\right)} \]

   2. add-sqr-sqrt 99.7%

      \[\leadsto R \cdot \color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}\right)}^{2}, \sqrt[3]{\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_2\right)}, \left(-\sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \]

9. Applied egg-rr 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 82.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.9e-11)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (*
    R
    (pow
     (sqrt (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- phi1 phi2)))
     2.0))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9e-11) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * pow(sqrt(hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2))), 2.0);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9e-11) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.pow(Math.sqrt(Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (phi1 - phi2))), 2.0);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.9e-11:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.pow(math.sqrt(math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (phi1 - phi2))), 2.0)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.9e-11)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * (sqrt(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), Float64(phi1 - phi2))) ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.9e-11)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * (sqrt(hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2))) ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.9e-11], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Power[N[Sqrt[N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.8999999999999999e-11

   1. Initial program 64.6%

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

   2. Taylor expanded in phi2 around 0 55.2%

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

      1. unpow2 55.2%

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

      2. unpow2 55.2%

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

      3. unpow2 55.2%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 55.2%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 79.1%

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

      6. *-commutative 79.1%

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

   4. Simplified 79.1%

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

   if 1.8999999999999999e-11 < phi2

   1. Initial program 53.0%

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

      1. add-sqr-sqrt 52.9%

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

      2. pow2 52.9%

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

   3. Applied egg-rr 93.7%

      \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]

   4. Taylor expanded in phi1 around 0 93.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)}\right)}^{2}\\ \end{array} \]

Alternative 3: 94.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ R \cdot {\left(\sqrt[3]{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right), \phi_1 - \phi_2\right)}\right)}^{3} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R
  (pow
   (cbrt
    (hypot (* (- lambda1 lambda2) (cos (* 0.5 (+ phi1 phi2)))) (- phi1 phi2)))
   3.0)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * pow(cbrt(hypot(((lambda1 - lambda2) * cos((0.5 * (phi1 + phi2)))), (phi1 - phi2))), 3.0);
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	return R * Math.pow(Math.cbrt(Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * (phi1 + phi2)))), (phi1 - phi2))), 3.0);
}

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(R * (cbrt(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * Float64(phi1 + phi2)))), Float64(phi1 - phi2))) ^ 3.0))
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Power[N[Power[N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

   1. add-cube-cbrt 60.9%

      \[\leadsto R \cdot \color{blue}{\left(\left(\sqrt[3]{\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}} \cdot \sqrt[3]{\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right) \cdot \sqrt[3]{\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right)} \]

   2. pow3 61.0%

      \[\leadsto R \cdot \color{blue}{{\left(\sqrt[3]{\sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}}\right)}^{3}} \]

3. Applied egg-rr 95.7%

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

4. Final simplification 95.7%

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

Alternative 4: 95.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Power[N[Sqrt[N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * N[(phi1 + phi2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5%

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

   1. add-sqr-sqrt 61.2%

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

   2. pow2 61.2%

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

3. Applied egg-rr 96.4%

   \[\leadsto R \cdot \color{blue}{{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]

4. Final simplification 96.4%

   \[\leadsto R \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

Alternative 5: 75.9% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.75 \cdot 10^{+119}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+159} \lor \neg \left(\phi_2 \leq 1.5 \cdot 10^{+173}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.75e+119)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (if (or (<= phi2 1.75e+159) (not (<= phi2 1.5e+173)))
     (* R (- phi2 phi1))
     (* R (* (- lambda1 lambda2) (cos (* 0.5 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.75e+119) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else if ((phi2 <= 1.75e+159) || !(phi2 <= 1.5e+173)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * ((lambda1 - lambda2) * cos((0.5 * phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.75e+119) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else if ((phi2 <= 1.75e+159) || !(phi2 <= 1.5e+173)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * ((lambda1 - lambda2) * Math.cos((0.5 * phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.75e+119:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	elif (phi2 <= 1.75e+159) or not (phi2 <= 1.5e+173):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * ((lambda1 - lambda2) * math.cos((0.5 * phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.75e+119)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	elseif ((phi2 <= 1.75e+159) || !(phi2 <= 1.5e+173))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.75e+119)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	elseif ((phi2 <= 1.75e+159) || ~((phi2 <= 1.5e+173)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * ((lambda1 - lambda2) * cos((0.5 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.75e+119], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[phi2, 1.75e+159], N[Not[LessEqual[phi2, 1.5e+173]], $MachinePrecision]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.75e119

   1. Initial program 63.6%

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

   2. Taylor expanded in phi2 around 0 52.4%

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

      1. unpow2 52.4%

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

      2. unpow2 52.4%

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

      3. unpow2 52.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 52.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 75.2%

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

      6. *-commutative 75.2%

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

   4. Simplified 75.2%

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

   if 1.75e119 < phi2 < 1.75e159 or 1.4999999999999999e173 < phi2

   1. Initial program 45.9%

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

   2. Taylor expanded in phi1 around -inf 79.1%

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

      1. +-commutative 79.1%

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

      2. mul-1-neg 79.1%

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

      3. unsub-neg 79.1%

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

   4. Simplified 79.1%

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

   if 1.75e159 < phi2 < 1.4999999999999999e173

   1. Initial program 72.6%

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

   2. Taylor expanded in lambda1 around inf 43.1%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1 + -1 \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 43.1%

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

      2. *-commutative 43.1%

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

      3. *-commutative 43.1%

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

      4. *-commutative 43.1%

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

      5. distribute-rgt-neg-out 43.1%

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

      6. distribute-lft-out 43.1%

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

      7. sub-neg 43.1%

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

      8. *-commutative 43.1%

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

      9. *-commutative 43.1%

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

   4. Simplified 43.1%

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

   5. Taylor expanded in phi1 around 0 43.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.75 \cdot 10^{+119}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+159} \lor \neg \left(\phi_2 \leq 1.5 \cdot 10^{+173}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 6: 79.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00026:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.00026)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* 0.5 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00026) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.00026) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((0.5 * phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.00026:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((0.5 * phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.00026)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.00026)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.00026], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.59999999999999977e-4

   1. Initial program 57.2%

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

   2. Taylor expanded in phi2 around 0 54.4%

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

      1. unpow2 54.4%

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

      2. unpow2 54.4%

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

      3. unpow2 54.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 54.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 81.6%

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

      6. *-commutative 81.6%

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

   4. Simplified 81.6%

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

   if -2.59999999999999977e-4 < phi1

   1. Initial program 62.9%

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

   2. Taylor expanded in phi1 around 0 57.1%

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

      1. unpow2 57.1%

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

      2. unpow2 57.1%

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

      3. unpow2 57.1%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 57.1%

         \[\leadsto R \cdot \sqrt{\phi_2 \cdot \phi_2 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 85.1%

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

   4. Simplified 85.1%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.00026:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 7: 71.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.5 \cdot 10^{+118}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+159} \lor \neg \left(\phi_2 \leq 1.5 \cdot 10^{+173}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.5e+118)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (or (<= phi2 1.75e+159) (not (<= phi2 1.5e+173)))
     (* R (- phi2 phi1))
     (* R (* (- lambda1 lambda2) (cos (* 0.5 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.5e+118) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if ((phi2 <= 1.75e+159) || !(phi2 <= 1.5e+173)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * ((lambda1 - lambda2) * cos((0.5 * phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.5e+118) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else if ((phi2 <= 1.75e+159) || !(phi2 <= 1.5e+173)) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * ((lambda1 - lambda2) * Math.cos((0.5 * phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.5e+118:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	elif (phi2 <= 1.75e+159) or not (phi2 <= 1.5e+173):
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * ((lambda1 - lambda2) * math.cos((0.5 * phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.5e+118)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif ((phi2 <= 1.75e+159) || !(phi2 <= 1.5e+173))
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.5e+118)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	elseif ((phi2 <= 1.75e+159) || ~((phi2 <= 1.5e+173)))
		tmp = R * (phi2 - phi1);
	else
		tmp = R * ((lambda1 - lambda2) * cos((0.5 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.5e+118], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[phi2, 1.75e+159], N[Not[LessEqual[phi2, 1.5e+173]], $MachinePrecision]], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 5.5000000000000003e118

   1. Initial program 63.6%

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

   2. Taylor expanded in phi2 around 0 52.4%

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

      1. unpow2 52.4%

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

      2. unpow2 52.4%

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

      3. unpow2 52.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 52.4%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 75.2%

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

      6. *-commutative 75.2%

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

   4. Simplified 75.2%

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

   5. Taylor expanded in phi1 around 0 69.4%

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

   if 5.5000000000000003e118 < phi2 < 1.75e159 or 1.4999999999999999e173 < phi2

   1. Initial program 45.9%

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

   2. Taylor expanded in phi1 around -inf 79.1%

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

      1. +-commutative 79.1%

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

      2. mul-1-neg 79.1%

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

      3. unsub-neg 79.1%

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

   4. Simplified 79.1%

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

   if 1.75e159 < phi2 < 1.4999999999999999e173

   1. Initial program 72.6%

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

   2. Taylor expanded in lambda1 around inf 43.1%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right) \cdot \lambda_1 + -1 \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   3. Step-by-step derivation

      1. mul-1-neg 43.1%

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

      2. *-commutative 43.1%

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

      3. *-commutative 43.1%

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

      4. *-commutative 43.1%

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

      5. distribute-rgt-neg-out 43.1%

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

      6. distribute-lft-out 43.1%

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

      7. sub-neg 43.1%

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

      8. *-commutative 43.1%

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

      9. *-commutative 43.1%

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

   4. Simplified 43.1%

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

   5. Taylor expanded in phi1 around 0 43.1%

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

4. Final simplification 70.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.5 \cdot 10^{+118}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+159} \lor \neg \left(\phi_2 \leq 1.5 \cdot 10^{+173}\right):\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 8: 71.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.5 \cdot 10^{+118} \lor \neg \left(\phi_2 \leq 2.25 \cdot 10^{+154}\right) \land \phi_2 \leq 7 \cdot 10^{+176}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (or (<= phi2 5.5e+118) (and (not (<= phi2 2.25e+154)) (<= phi2 7e+176)))
   (* 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 <= 5.5e+118) || (!(phi2 <= 2.25e+154) && (phi2 <= 7e+176))) {
		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 <= 5.5e+118) || (!(phi2 <= 2.25e+154) && (phi2 <= 7e+176))) {
		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 <= 5.5e+118) or (not (phi2 <= 2.25e+154) and (phi2 <= 7e+176)):
		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 <= 5.5e+118) || (!(phi2 <= 2.25e+154) && (phi2 <= 7e+176)))
		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 <= 5.5e+118) || (~((phi2 <= 2.25e+154)) && (phi2 <= 7e+176)))
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[Or[LessEqual[phi2, 5.5e+118], And[N[Not[LessEqual[phi2, 2.25e+154]], $MachinePrecision], LessEqual[phi2, 7e+176]]], 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 < 5.5000000000000003e118 or 2.25000000000000005e154 < phi2 < 7.00000000000000005e176

   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. Taylor expanded in phi2 around 0 53.3%

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

      1. unpow2 53.3%

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

      2. unpow2 53.3%

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

      3. unpow2 53.3%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 53.3%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 75.6%

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

      6. *-commutative 75.6%

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

   4. Simplified 75.6%

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

   5. Taylor expanded in phi1 around 0 70.1%

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

   if 5.5000000000000003e118 < phi2 < 2.25000000000000005e154 or 7.00000000000000005e176 < phi2

   1. Initial program 43.7%

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

   2. Taylor expanded in phi1 around -inf 80.1%

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

      1. +-commutative 80.1%

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

      2. mul-1-neg 80.1%

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

      3. unsub-neg 80.1%

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

   4. Simplified 80.1%

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

4. Final simplification 71.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.5 \cdot 10^{+118} \lor \neg \left(\phi_2 \leq 2.25 \cdot 10^{+154}\right) \land \phi_2 \leq 7 \cdot 10^{+176}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 9: 44.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+128}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.25 \cdot 10^{-252}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, -\lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.5e+128)
   (* R (- lambda1))
   (if (<= lambda1 2.25e-252)
     (* R (- phi2 phi1))
     (* R (hypot phi1 (- lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e+128) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 2.25e-252) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * hypot(phi1, -lambda2);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.5e+128) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 2.25e-252) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * Math.hypot(phi1, -lambda2);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.5e+128:
		tmp = R * -lambda1
	elif lambda1 <= 2.25e-252:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * math.hypot(phi1, -lambda2)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.5e+128)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda1 <= 2.25e-252)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * hypot(phi1, Float64(-lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.5e+128)
		tmp = R * -lambda1;
	elseif (lambda1 <= 2.25e-252)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * hypot(phi1, -lambda2);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.5e+128], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, 2.25e-252], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi1 ^ 2 + (-lambda2) ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.4999999999999999e128

   1. Initial program 53.1%

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

   2. Taylor expanded in phi2 around 0 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 51.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 79.9%

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

      6. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in lambda1 around -inf 64.2%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

         \[\leadsto R \cdot \color{blue}{\left(-\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)} \]

      2. distribute-rgt-neg-in 64.2%

         \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)} \]

   7. Simplified 64.2%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)} \]

   8. Taylor expanded in phi1 around 0 64.8%

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

      1. mul-1-neg 64.8%

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

      2. *-commutative 64.8%

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

      3. distribute-rgt-neg-in 64.8%

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

   10. Simplified 64.8%

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

   if -1.4999999999999999e128 < lambda1 < 2.2500000000000001e-252

   1. Initial program 61.0%

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

   2. Taylor expanded in phi1 around -inf 37.3%

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

      1. +-commutative 37.3%

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

      2. mul-1-neg 37.3%

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

      3. unsub-neg 37.3%

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

   4. Simplified 37.3%

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

   if 2.2500000000000001e-252 < lambda1

   1. Initial program 65.0%

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

   2. Taylor expanded in phi2 around 0 51.9%

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

      1. unpow2 51.9%

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

      2. unpow2 51.9%

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

      3. unpow2 51.9%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 51.9%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 75.5%

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

      6. *-commutative 75.5%

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

   4. Simplified 75.5%

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

   5. Taylor expanded in lambda1 around 0 57.6%

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

      1. mul-1-neg 57.6%

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

      2. *-commutative 57.6%

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

      3. distribute-rgt-neg-in 57.6%

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

   7. Simplified 57.6%

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

   8. Taylor expanded in phi1 around 0 53.6%

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

      1. neg-mul-1 53.6%

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

   10. Simplified 53.6%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.5 \cdot 10^{+128}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.25 \cdot 10^{-252}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, -\lambda_2\right)\\ \end{array} \]

Alternative 10: 28.6% accurate, 36.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.3 \cdot 10^{-231}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.8e+131)
   (* R (- lambda1))
   (if (<= lambda1 2.3e-231) (* R (- phi2 phi1)) (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.8e+131) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 2.3e-231) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * lambda2;
	}
	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 <= (-2.8d+131)) then
        tmp = r * -lambda1
    else if (lambda1 <= 2.3d-231) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * lambda2
    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 <= -2.8e+131) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 2.3e-231) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.8e+131:
		tmp = R * -lambda1
	elif lambda1 <= 2.3e-231:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.8e+131)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda1 <= 2.3e-231)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.8e+131)
		tmp = R * -lambda1;
	elseif (lambda1 <= 2.3e-231)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.8e+131], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, 2.3e-231], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -2.8000000000000001e131

   1. Initial program 53.1%

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

   2. Taylor expanded in phi2 around 0 51.1%

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

      1. unpow2 51.1%

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

      2. unpow2 51.1%

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

      3. unpow2 51.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 51.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 79.9%

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

      6. *-commutative 79.9%

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

   4. Simplified 79.9%

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

   5. Taylor expanded in lambda1 around -inf 64.2%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 64.2%

         \[\leadsto R \cdot \color{blue}{\left(-\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)} \]

      2. distribute-rgt-neg-in 64.2%

         \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)} \]

   7. Simplified 64.2%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)} \]

   8. Taylor expanded in phi1 around 0 64.8%

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

      1. mul-1-neg 64.8%

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

      2. *-commutative 64.8%

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

      3. distribute-rgt-neg-in 64.8%

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

   10. Simplified 64.8%

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

   if -2.8000000000000001e131 < lambda1 < 2.3e-231

   1. Initial program 62.1%

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

   2. Taylor expanded in phi1 around -inf 38.1%

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

      1. +-commutative 38.1%

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

      2. mul-1-neg 38.1%

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

      3. unsub-neg 38.1%

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

   4. Simplified 38.1%

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

   if 2.3e-231 < lambda1

   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. Taylor expanded in phi2 around 0 51.5%

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

      1. unpow2 51.5%

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

      2. unpow2 51.5%

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

      3. unpow2 51.5%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 51.5%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 75.7%

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

      6. *-commutative 75.7%

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

   4. Simplified 75.7%

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

   5. Taylor expanded in lambda1 around 0 57.3%

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

      1. mul-1-neg 57.3%

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

      2. *-commutative 57.3%

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

      3. distribute-rgt-neg-in 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in phi1 around 0 15.9%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.8 \cdot 10^{+131}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 2.3 \cdot 10^{-231}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]

Alternative 11: 28.1% accurate, 40.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.8 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.55 \cdot 10^{-307}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -7.8e-14)
   (* R (- phi1))
   (if (<= phi1 -1.55e-307) (* R (- lambda1)) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.8e-14) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.55e-307) {
		tmp = R * -lambda1;
	} 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 <= (-7.8d-14)) then
        tmp = r * -phi1
    else if (phi1 <= (-1.55d-307)) then
        tmp = r * -lambda1
    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 <= -7.8e-14) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.55e-307) {
		tmp = R * -lambda1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -7.8e-14:
		tmp = R * -phi1
	elif phi1 <= -1.55e-307:
		tmp = R * -lambda1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -7.8e-14)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -1.55e-307)
		tmp = Float64(R * Float64(-lambda1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -7.8e-14)
		tmp = R * -phi1;
	elseif (phi1 <= -1.55e-307)
		tmp = R * -lambda1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -7.8e-14], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -1.55e-307], N[(R * (-lambda1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -7.7999999999999996e-14

   1. Initial program 57.0%

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

   2. Taylor expanded in phi1 around -inf 58.6%

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

      1. mul-1-neg 58.6%

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

   4. Simplified 58.6%

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

   if -7.7999999999999996e-14 < phi1 < -1.5499999999999999e-307

   1. Initial program 62.0%

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

   2. Taylor expanded in phi2 around 0 42.8%

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

      1. unpow2 42.8%

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

      2. unpow2 42.8%

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

      3. unpow2 42.8%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 42.8%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 56.7%

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

      6. *-commutative 56.7%

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

   4. Simplified 56.7%

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

   5. Taylor expanded in lambda1 around -inf 21.6%

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg 21.6%

         \[\leadsto R \cdot \color{blue}{\left(-\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1\right)} \]

      2. distribute-rgt-neg-in 21.6%

         \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)} \]

   7. Simplified 21.6%

      \[\leadsto R \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_1\right)\right)} \]

   8. Taylor expanded in phi1 around 0 21.6%

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

      1. mul-1-neg 21.6%

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

      2. *-commutative 21.6%

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

      3. distribute-rgt-neg-in 21.6%

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

   10. Simplified 21.6%

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

   if -1.5499999999999999e-307 < phi1

   1. Initial program 63.6%

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

   2. Taylor expanded in phi2 around inf 17.4%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.8 \cdot 10^{-14}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.55 \cdot 10^{-307}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 12: 28.3% accurate, 46.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.5 \cdot 10^{-209}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 410:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.5e-209)
   (* R (- phi1))
   (if (<= phi2 410.0) (* R lambda2) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.5e-209) {
		tmp = R * -phi1;
	} else if (phi2 <= 410.0) {
		tmp = R * lambda2;
	} 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 <= 4.5d-209) then
        tmp = r * -phi1
    else if (phi2 <= 410.0d0) then
        tmp = r * lambda2
    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 <= 4.5e-209) {
		tmp = R * -phi1;
	} else if (phi2 <= 410.0) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.5e-209:
		tmp = R * -phi1
	elif phi2 <= 410.0:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.5e-209)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi2 <= 410.0)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.5e-209)
		tmp = R * -phi1;
	elseif (phi2 <= 410.0)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.5e-209], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi2, 410.0], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 4.4999999999999998e-209

   1. Initial program 66.0%

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

   2. Taylor expanded in phi1 around -inf 17.7%

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

      1. mul-1-neg 17.7%

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

   4. Simplified 17.7%

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

   if 4.4999999999999998e-209 < phi2 < 410

   1. Initial program 59.7%

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

   2. Taylor expanded in phi2 around 0 59.7%

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

      1. unpow2 59.7%

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

      2. unpow2 59.7%

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

      3. unpow2 59.7%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 59.7%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 97.7%

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

      6. *-commutative 97.7%

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

   4. Simplified 97.7%

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

   5. Taylor expanded in lambda1 around 0 64.5%

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

      1. mul-1-neg 64.5%

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

      2. *-commutative 64.5%

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

      3. distribute-rgt-neg-in 64.5%

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

   7. Simplified 64.5%

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

   8. Taylor expanded in phi1 around 0 20.8%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]

   if 410 < phi2

   1. Initial program 53.0%

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

   2. Taylor expanded in phi2 around inf 52.4%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.5 \cdot 10^{-209}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_2 \leq 410:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 13: 25.6% accurate, 65.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 9.5e+32) (* R phi2) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 9.5e+32) {
		tmp = R * phi2;
	} else {
		tmp = R * lambda2;
	}
	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 (lambda2 <= 9.5d+32) then
        tmp = r * phi2
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 9.5e+32) {
		tmp = R * phi2;
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 9.5e+32:
		tmp = R * phi2
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 9.5e+32)
		tmp = Float64(R * phi2);
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 9.5e+32)
		tmp = R * phi2;
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 9.5e+32], N[(R * phi2), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.50000000000000006e32

   1. Initial program 60.8%

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

   2. Taylor expanded in phi2 around inf 18.5%

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

   if 9.50000000000000006e32 < lambda2

   1. Initial program 63.9%

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

   2. Taylor expanded in phi2 around 0 61.1%

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

      1. unpow2 61.1%

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

      2. unpow2 61.1%

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

      3. unpow2 61.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \color{blue}{\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      4. unswap-sqr 61.1%

         \[\leadsto R \cdot \sqrt{\phi_1 \cdot \phi_1 + \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}} \]

      5. hypot-def 78.9%

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

      6. *-commutative 78.9%

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

   4. Simplified 78.9%

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

   5. Taylor expanded in lambda1 around 0 72.1%

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

      1. mul-1-neg 72.1%

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

      2. *-commutative 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

   7. Simplified 72.1%

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

   8. Taylor expanded in phi1 around 0 63.5%

      \[\leadsto \color{blue}{\lambda_2 \cdot R} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.5 \cdot 10^{+32}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]

Alternative 14: 17.7% 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 61.5%

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

2. Taylor expanded in phi2 around inf 16.7%

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

3. Final simplification 16.7%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023187 
(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))))))