Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.7% → 99.9%

Time: 51.9s

Alternatives: 15

Speedup: 3.0×

• 30.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\\ R \cdot \sqrt{t\_0 \cdot t\_0 + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. hypot-define 96.9%

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

3. Simplified 96.9%

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

   1. add-cube-cbrt 96.6%

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

   2. pow2 96.6%

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

   3. div-inv 96.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right)}^{2} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right), \phi_1 - \phi_2\right) \]

   4. metadata-eval 96.6%

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

   5. div-inv 96.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}^{2} \cdot \sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

   6. metadata-eval 96.6%

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

6. Applied egg-rr 96.6%

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

   1. *-commutative 96.6%

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

   2. +-commutative 96.6%

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

   3. distribute-lft-in 96.6%

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

   4. cos-sum 96.7%

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

   5. *-commutative 96.7%

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

   6. *-commutative 96.7%

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

   7. *-commutative 96.7%

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

   8. *-commutative 96.7%

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

8. Applied egg-rr 96.7%

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

   1. *-commutative 96.6%

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

   2. +-commutative 96.6%

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

   3. distribute-lft-in 96.6%

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

   4. cos-sum 96.7%

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

   5. *-commutative 96.7%

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

   6. *-commutative 96.7%

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

   7. *-commutative 96.7%

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

   8. *-commutative 96.7%

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

10. Applied egg-rr 99.5%

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

    1. unpow2 99.5%

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

    2. add-cube-cbrt 99.9%

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

    3. cancel-sign-sub-inv 99.9%

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

    4. fma-define 99.9%

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

    5. *-commutative 99.9%

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

    6. *-commutative 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. hypot-define 96.9%

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

3. Simplified 96.9%

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

   1. add-cube-cbrt 96.6%

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

   2. pow2 96.6%

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

   3. div-inv 96.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right)}^{2} \cdot \sqrt[3]{\cos \left(\frac{\phi_1 + \phi_2}{2}\right)}\right), \phi_1 - \phi_2\right) \]

   4. metadata-eval 96.6%

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

   5. div-inv 96.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left({\left(\sqrt[3]{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right)}\right)}^{2} \cdot \sqrt[3]{\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}}\right), \phi_1 - \phi_2\right) \]

   6. metadata-eval 96.6%

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

6. Applied egg-rr 96.6%

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

   1. *-commutative 96.6%

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

   2. +-commutative 96.6%

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

   3. distribute-lft-in 96.6%

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

   4. cos-sum 96.7%

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

   5. *-commutative 96.7%

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

   6. *-commutative 96.7%

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

   7. *-commutative 96.7%

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

   8. *-commutative 96.7%

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

8. Applied egg-rr 96.7%

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

   1. *-commutative 96.6%

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

   2. +-commutative 96.6%

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

   3. distribute-lft-in 96.6%

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

   4. cos-sum 96.7%

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

   5. *-commutative 96.7%

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

   6. *-commutative 96.7%

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

   7. *-commutative 96.7%

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

   8. *-commutative 96.7%

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

10. Applied egg-rr 99.5%

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

    1. unpow2 99.5%

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

    2. add-cube-cbrt 99.9%

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

    3. *-commutative 99.9%

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

    4. *-commutative 99.9%

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

12. Applied egg-rr 99.9%

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

13. Final simplification 99.9%

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

Alternative 3: 76.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.1 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \cos \left(\phi_2 \cdot 0.5\right) \cdot \left(-\lambda_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5.1e+20)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (hypot phi2 (* (cos (* phi2 0.5)) (- lambda2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.1e+20) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * hypot(phi2, (cos((phi2 * 0.5)) * -lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.1e+20) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * Math.hypot(phi2, (Math.cos((phi2 * 0.5)) * -lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.1e+20:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * math.hypot(phi2, (math.cos((phi2 * 0.5)) * -lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.1e+20)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(cos(Float64(phi2 * 0.5)) * Float64(-lambda2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.1e+20)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * hypot(phi2, (cos((phi2 * 0.5)) * -lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5.1e+20], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * (-lambda2)), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.1e20

   1. Initial program 57.3%

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

      1. hypot-define 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in phi2 around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. unpow2 45.3%

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

      4. unpow2 45.3%

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

      5. unswap-sqr 45.3%

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

      6. hypot-define 73.9%

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

   7. Simplified 73.9%

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

   if 5.1e20 < phi2

   1. Initial program 50.3%

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

      1. hypot-define 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in phi1 around 0 49.3%

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

      1. +-commutative 49.3%

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

      2. unpow2 49.3%

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

      3. unpow2 49.3%

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

      4. unpow2 49.3%

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

      5. unswap-sqr 49.3%

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

      6. hypot-define 91.7%

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

   7. Simplified 91.7%

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

   8. Taylor expanded in lambda1 around 0 74.3%

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

      1. mul-1-neg 74.3%

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

      2. *-commutative 74.3%

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

      3. *-commutative 74.3%

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

      4. distribute-rgt-neg-out 74.3%

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

      5. *-commutative 74.3%

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

   10. Simplified 74.3%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.1 \cdot 10^{+20}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \cos \left(\phi_2 \cdot 0.5\right) \cdot \left(-\lambda_2\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -76000000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -76000000000.0)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -76000000000.0) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -76000000000.0) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -76000000000.0:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -76000000000.0)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -76000000000.0)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -76000000000.0], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.6e10

   1. Initial program 48.9%

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

      1. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in phi2 around 0 45.3%

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

      1. +-commutative 45.3%

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

      2. unpow2 45.3%

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

      3. unpow2 45.3%

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

      4. unpow2 45.3%

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

      5. unswap-sqr 45.3%

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

      6. hypot-define 80.8%

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

   7. Simplified 80.8%

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

   if -7.6e10 < phi1

   1. Initial program 58.4%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in phi1 around 0 49.8%

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

      1. +-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. unpow2 49.8%

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

      5. unswap-sqr 49.8%

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

      6. hypot-define 79.5%

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

   7. Simplified 79.5%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -76000000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

   1. hypot-define 96.9%

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

3. Simplified 96.9%

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

5. Final simplification 96.9%

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

Alternative 6: 70.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -205000000000:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -205000000000.0)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -205000000000.0) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -205000000000.0) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -205000000000.0:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -205000000000.0)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -205000000000.0)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -205000000000.0], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.05e11

   1. Initial program 48.9%

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

      1. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in phi1 around -inf 52.7%

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

      1. associate-*r* 52.7%

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

      2. mul-1-neg 52.7%

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

      3. mul-1-neg 52.7%

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

      4. unsub-neg 52.7%

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

      5. associate-/l* 55.6%

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

   7. Simplified 55.6%

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

   if -2.05e11 < phi1

   1. Initial program 58.4%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in phi1 around 0 49.8%

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

      1. +-commutative 49.8%

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

      2. unpow2 49.8%

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

      3. unpow2 49.8%

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

      4. unpow2 49.8%

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

      5. unswap-sqr 49.8%

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

      6. hypot-define 79.5%

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

   7. Simplified 79.5%

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

   8. Taylor expanded in phi2 around 0 70.0%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -205000000000:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.0% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -160000000000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -3.05 \cdot 10^{-143} \lor \neg \left(\phi_1 \leq -1.05 \cdot 10^{-177}\right) \land \phi_1 \leq 4.4 \cdot 10^{-211}:\\ \;\;\;\;\lambda_2 \cdot \left(R - R \cdot \frac{\lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -160000000000.0)
   (* phi1 (- R))
   (if (or (<= phi1 -3.05e-143)
           (and (not (<= phi1 -1.05e-177)) (<= phi1 4.4e-211)))
     (* lambda2 (- R (* R (/ lambda1 lambda2))))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -160000000000.0) {
		tmp = phi1 * -R;
	} else if ((phi1 <= -3.05e-143) || (!(phi1 <= -1.05e-177) && (phi1 <= 4.4e-211))) {
		tmp = lambda2 * (R - (R * (lambda1 / 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 (phi1 <= (-160000000000.0d0)) then
        tmp = phi1 * -r
    else if ((phi1 <= (-3.05d-143)) .or. (.not. (phi1 <= (-1.05d-177))) .and. (phi1 <= 4.4d-211)) then
        tmp = lambda2 * (r - (r * (lambda1 / 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 (phi1 <= -160000000000.0) {
		tmp = phi1 * -R;
	} else if ((phi1 <= -3.05e-143) || (!(phi1 <= -1.05e-177) && (phi1 <= 4.4e-211))) {
		tmp = lambda2 * (R - (R * (lambda1 / lambda2)));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -160000000000.0:
		tmp = phi1 * -R
	elif (phi1 <= -3.05e-143) or (not (phi1 <= -1.05e-177) and (phi1 <= 4.4e-211)):
		tmp = lambda2 * (R - (R * (lambda1 / lambda2)))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -160000000000.0)
		tmp = Float64(phi1 * Float64(-R));
	elseif ((phi1 <= -3.05e-143) || (!(phi1 <= -1.05e-177) && (phi1 <= 4.4e-211)))
		tmp = Float64(lambda2 * Float64(R - Float64(R * Float64(lambda1 / lambda2))));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -160000000000.0)
		tmp = phi1 * -R;
	elseif ((phi1 <= -3.05e-143) || (~((phi1 <= -1.05e-177)) && (phi1 <= 4.4e-211)))
		tmp = lambda2 * (R - (R * (lambda1 / lambda2)));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -160000000000.0], N[(phi1 * (-R)), $MachinePrecision], If[Or[LessEqual[phi1, -3.05e-143], And[N[Not[LessEqual[phi1, -1.05e-177]], $MachinePrecision], LessEqual[phi1, 4.4e-211]]], N[(lambda2 * N[(R - N[(R * N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.6e11

   1. Initial program 48.9%

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

      1. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in phi1 around -inf 55.8%

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

      1. mul-1-neg 55.8%

         \[\leadsto \color{blue}{-R \cdot \phi_1} \]

      2. *-commutative 55.8%

         \[\leadsto -\color{blue}{\phi_1 \cdot R} \]

      3. distribute-rgt-neg-in 55.8%

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

   7. Simplified 55.8%

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

   if -1.6e11 < phi1 < -3.04999999999999996e-143 or -1.05e-177 < phi1 < 4.39999999999999996e-211

   1. Initial program 58.4%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi1 around 0 55.4%

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

      1. +-commutative 55.4%

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

      2. unpow2 55.4%

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

      3. unpow2 55.4%

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

      4. unpow2 55.4%

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

      5. unswap-sqr 55.4%

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

      6. hypot-define 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in lambda2 around inf 31.4%

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

      1. +-commutative 31.4%

         \[\leadsto \lambda_2 \cdot \color{blue}{\left(R \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}{\lambda_2}\right)} \]

      2. mul-1-neg 31.4%

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

      3. unsub-neg 31.4%

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

      4. *-commutative 31.4%

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

      5. associate-/l* 27.8%

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

      6. *-commutative 27.8%

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

   10. Simplified 27.8%

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

   11. Taylor expanded in phi2 around 0 27.0%

       \[\leadsto \color{blue}{\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)} \]
   12. Step-by-step derivation

       1. associate-/l* 23.4%

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

   13. Simplified 23.4%

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

   if -3.04999999999999996e-143 < phi1 < -1.05e-177 or 4.39999999999999996e-211 < phi1

   1. Initial program 58.5%

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

      1. hypot-define 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in phi2 around inf 18.2%

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

      1. *-commutative 18.2%

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

   7. Simplified 18.2%

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

4. Final simplification 29.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -160000000000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -3.05 \cdot 10^{-143} \lor \neg \left(\phi_1 \leq -1.05 \cdot 10^{-177}\right) \land \phi_1 \leq 4.4 \cdot 10^{-211}:\\ \;\;\;\;\lambda_2 \cdot \left(R - R \cdot \frac{\lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 31.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -210000000000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-143} \lor \neg \left(\phi_1 \leq -1.75 \cdot 10^{-177}\right) \land \phi_1 \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -210000000000.0)
   (* phi1 (- R))
   (if (or (<= phi1 -3e-143) (and (not (<= phi1 -1.75e-177)) (<= phi1 5e-211)))
     (* lambda2 (- R (/ (* R lambda1) lambda2)))
     (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -210000000000.0) {
		tmp = phi1 * -R;
	} else if ((phi1 <= -3e-143) || (!(phi1 <= -1.75e-177) && (phi1 <= 5e-211))) {
		tmp = lambda2 * (R - ((R * lambda1) / 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 (phi1 <= (-210000000000.0d0)) then
        tmp = phi1 * -r
    else if ((phi1 <= (-3d-143)) .or. (.not. (phi1 <= (-1.75d-177))) .and. (phi1 <= 5d-211)) then
        tmp = lambda2 * (r - ((r * lambda1) / 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 (phi1 <= -210000000000.0) {
		tmp = phi1 * -R;
	} else if ((phi1 <= -3e-143) || (!(phi1 <= -1.75e-177) && (phi1 <= 5e-211))) {
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -210000000000.0:
		tmp = phi1 * -R
	elif (phi1 <= -3e-143) or (not (phi1 <= -1.75e-177) and (phi1 <= 5e-211)):
		tmp = lambda2 * (R - ((R * lambda1) / lambda2))
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -210000000000.0)
		tmp = Float64(phi1 * Float64(-R));
	elseif ((phi1 <= -3e-143) || (!(phi1 <= -1.75e-177) && (phi1 <= 5e-211)))
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -210000000000.0)
		tmp = phi1 * -R;
	elseif ((phi1 <= -3e-143) || (~((phi1 <= -1.75e-177)) && (phi1 <= 5e-211)))
		tmp = lambda2 * (R - ((R * lambda1) / lambda2));
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -210000000000.0], N[(phi1 * (-R)), $MachinePrecision], If[Or[LessEqual[phi1, -3e-143], And[N[Not[LessEqual[phi1, -1.75e-177]], $MachinePrecision], LessEqual[phi1, 5e-211]]], N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -2.1e11

   1. Initial program 48.9%

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

      1. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in phi1 around -inf 55.8%

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

      1. mul-1-neg 55.8%

         \[\leadsto \color{blue}{-R \cdot \phi_1} \]

      2. *-commutative 55.8%

         \[\leadsto -\color{blue}{\phi_1 \cdot R} \]

      3. distribute-rgt-neg-in 55.8%

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

   7. Simplified 55.8%

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

   if -2.1e11 < phi1 < -2.99999999999999985e-143 or -1.7500000000000001e-177 < phi1 < 5.0000000000000002e-211

   1. Initial program 58.4%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi1 around 0 55.4%

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

      1. +-commutative 55.4%

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

      2. unpow2 55.4%

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

      3. unpow2 55.4%

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

      4. unpow2 55.4%

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

      5. unswap-sqr 55.4%

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

      6. hypot-define 95.5%

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

   7. Simplified 95.5%

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

   8. Taylor expanded in lambda2 around inf 31.4%

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

      1. +-commutative 31.4%

         \[\leadsto \lambda_2 \cdot \color{blue}{\left(R \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}{\lambda_2}\right)} \]

      2. mul-1-neg 31.4%

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

      3. unsub-neg 31.4%

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

      4. *-commutative 31.4%

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

      5. associate-/l* 27.8%

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

      6. *-commutative 27.8%

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

   10. Simplified 27.8%

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

   11. Taylor expanded in phi2 around 0 27.0%

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

   if -2.99999999999999985e-143 < phi1 < -1.7500000000000001e-177 or 5.0000000000000002e-211 < phi1

   1. Initial program 58.5%

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

      1. hypot-define 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in phi2 around inf 18.2%

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

      1. *-commutative 18.2%

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

   7. Simplified 18.2%

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

4. Final simplification 30.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -210000000000:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-143} \lor \neg \left(\phi_1 \leq -1.75 \cdot 10^{-177}\right) \land \phi_1 \leq 5 \cdot 10^{-211}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.3% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{-25}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq -1.8 \cdot 10^{-177}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 6 \cdot 10^{-211}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* lambda2 (- R (/ (* R lambda1) lambda2)))))
   (if (<= phi1 -6.2e-25)
     (* phi1 (- (* R (/ phi2 phi1)) R))
     (if (<= phi1 -6.5e-143)
       t_0
       (if (<= phi1 -1.8e-177)
         (* phi2 (- R (* R (/ phi1 phi2))))
         (if (<= phi1 6e-211) t_0 (* R phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda2 * (R - ((R * lambda1) / lambda2));
	double tmp;
	if (phi1 <= -6.2e-25) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -6.5e-143) {
		tmp = t_0;
	} else if (phi1 <= -1.8e-177) {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	} else if (phi1 <= 6e-211) {
		tmp = t_0;
	} 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) :: t_0
    real(8) :: tmp
    t_0 = lambda2 * (r - ((r * lambda1) / lambda2))
    if (phi1 <= (-6.2d-25)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-6.5d-143)) then
        tmp = t_0
    else if (phi1 <= (-1.8d-177)) then
        tmp = phi2 * (r - (r * (phi1 / phi2)))
    else if (phi1 <= 6d-211) then
        tmp = t_0
    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 t_0 = lambda2 * (R - ((R * lambda1) / lambda2));
	double tmp;
	if (phi1 <= -6.2e-25) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -6.5e-143) {
		tmp = t_0;
	} else if (phi1 <= -1.8e-177) {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	} else if (phi1 <= 6e-211) {
		tmp = t_0;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = lambda2 * (R - ((R * lambda1) / lambda2))
	tmp = 0
	if phi1 <= -6.2e-25:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -6.5e-143:
		tmp = t_0
	elif phi1 <= -1.8e-177:
		tmp = phi2 * (R - (R * (phi1 / phi2)))
	elif phi1 <= 6e-211:
		tmp = t_0
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(lambda2 * Float64(R - Float64(Float64(R * lambda1) / lambda2)))
	tmp = 0.0
	if (phi1 <= -6.2e-25)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -6.5e-143)
		tmp = t_0;
	elseif (phi1 <= -1.8e-177)
		tmp = Float64(phi2 * Float64(R - Float64(R * Float64(phi1 / phi2))));
	elseif (phi1 <= 6e-211)
		tmp = t_0;
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = lambda2 * (R - ((R * lambda1) / lambda2));
	tmp = 0.0;
	if (phi1 <= -6.2e-25)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -6.5e-143)
		tmp = t_0;
	elseif (phi1 <= -1.8e-177)
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	elseif (phi1 <= 6e-211)
		tmp = t_0;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(lambda2 * N[(R - N[(N[(R * lambda1), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -6.2e-25], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -6.5e-143], t$95$0, If[LessEqual[phi1, -1.8e-177], N[(phi2 * N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 6e-211], t$95$0, N[(R * phi2), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -6.19999999999999989e-25

   1. Initial program 53.2%

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

      1. hypot-define 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in phi1 around -inf 49.5%

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

      1. associate-*r* 49.5%

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

      2. mul-1-neg 49.5%

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

      3. mul-1-neg 49.5%

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

      4. unsub-neg 49.5%

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

      5. associate-/l* 52.1%

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

   7. Simplified 52.1%

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

   if -6.19999999999999989e-25 < phi1 < -6.4999999999999999e-143 or -1.79999999999999991e-177 < phi1 < 6.00000000000000009e-211

   1. Initial program 55.1%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi1 around 0 55.1%

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

      1. +-commutative 55.1%

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

      2. unpow2 55.1%

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

      3. unpow2 55.1%

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

      4. unpow2 55.1%

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

      5. unswap-sqr 55.1%

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

      6. hypot-define 99.9%

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

   7. Simplified 99.9%

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

   8. Taylor expanded in lambda2 around inf 34.3%

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

      1. +-commutative 34.3%

         \[\leadsto \lambda_2 \cdot \color{blue}{\left(R \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}{\lambda_2}\right)} \]

      2. mul-1-neg 34.3%

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

      3. unsub-neg 34.3%

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

      4. *-commutative 34.3%

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

      5. associate-/l* 30.1%

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

      6. *-commutative 30.1%

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

   10. Simplified 30.1%

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

   11. Taylor expanded in phi2 around 0 29.0%

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

   if -6.4999999999999999e-143 < phi1 < -1.79999999999999991e-177

   1. Initial program 72.4%

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

      1. hypot-define 100.0%

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

   3. Simplified 100.0%

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

      1. expm1-log1p-u 56.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right)\right)} \]

      2. expm1-undefine 33.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right)} - 1} \]

      3. *-commutative 33.2%

         \[\leadsto e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right)\right)} - 1 \]

      4. div-inv 33.2%

         \[\leadsto e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1 \]

      5. metadata-eval 33.2%

         \[\leadsto e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1 \]

   6. Applied egg-rr 33.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]

   8. Simplified 56.2%

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

   9. Taylor expanded in phi2 around inf 30.5%

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

       1. mul-1-neg 30.5%

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

       2. unsub-neg 30.5%

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

       3. associate-/l* 30.5%

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

   11. Simplified 30.5%

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

   if 6.00000000000000009e-211 < phi1

   1. Initial program 57.5%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi2 around inf 18.1%

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

      1. *-commutative 18.1%

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

   7. Simplified 18.1%

      \[\leadsto \color{blue}{\phi_2 \cdot R} \]
3. Recombined 4 regimes into one program.

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{-25}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{elif}\;\phi_1 \leq -1.8 \cdot 10^{-177}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 6 \cdot 10^{-211}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{R \cdot \lambda_1}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 29.8% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.1 \cdot 10^{+158}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.1e+158)
   (* lambda1 (- R))
   (if (<= lambda1 1.85e-50)
     (* phi2 (- R (* R (/ phi1 phi2))))
     (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.1e+158) {
		tmp = lambda1 * -R;
	} else if (lambda1 <= 1.85e-50) {
		tmp = phi2 * (R - (R * (phi1 / 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 (lambda1 <= (-1.1d+158)) then
        tmp = lambda1 * -r
    else if (lambda1 <= 1.85d-50) then
        tmp = phi2 * (r - (r * (phi1 / 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 (lambda1 <= -1.1e+158) {
		tmp = lambda1 * -R;
	} else if (lambda1 <= 1.85e-50) {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.1e+158:
		tmp = lambda1 * -R
	elif lambda1 <= 1.85e-50:
		tmp = phi2 * (R - (R * (phi1 / phi2)))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.1e+158)
		tmp = Float64(lambda1 * Float64(-R));
	elseif (lambda1 <= 1.85e-50)
		tmp = Float64(phi2 * Float64(R - Float64(R * Float64(phi1 / phi2))));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.1e+158)
		tmp = lambda1 * -R;
	elseif (lambda1 <= 1.85e-50)
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.1e+158], N[(lambda1 * (-R)), $MachinePrecision], If[LessEqual[lambda1, 1.85e-50], N[(phi2 * N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.1000000000000001e158

   1. Initial program 40.7%

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

      1. hypot-define 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in phi1 around 0 40.7%

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

      1. +-commutative 40.7%

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

      2. unpow2 40.7%

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

      3. unpow2 40.7%

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

      4. unpow2 40.7%

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

      5. unswap-sqr 40.7%

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

      6. hypot-define 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in lambda2 around inf 45.3%

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

      1. +-commutative 45.3%

         \[\leadsto \lambda_2 \cdot \color{blue}{\left(R \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}{\lambda_2}\right)} \]

      2. mul-1-neg 45.3%

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

      3. unsub-neg 45.3%

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

      4. *-commutative 45.3%

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

      5. associate-/l* 35.1%

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

      6. *-commutative 35.1%

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

   10. Simplified 35.1%

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

   11. Taylor expanded in lambda2 around 0 45.7%

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

       1. *-commutative 45.7%

          \[\leadsto -1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)\right) \]

       2. *-commutative 45.7%

          \[\leadsto -1 \cdot \left(R \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_1\right)}\right) \]

       3. neg-mul-1 45.7%

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

       4. distribute-rgt-neg-in 45.7%

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

       5. *-commutative 45.7%

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

       6. distribute-rgt-neg-in 45.7%

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

   13. Simplified 45.7%

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

   14. Taylor expanded in phi2 around 0 57.5%

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

       1. mul-1-neg 57.5%

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

       2. distribute-rgt-neg-in 57.5%

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

   16. Simplified 57.5%

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

   if -1.1000000000000001e158 < lambda1 < 1.85e-50

   1. Initial program 63.0%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

      1. expm1-log1p-u 66.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right)\right)} \]

      2. expm1-undefine 43.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right)} - 1} \]

      3. *-commutative 43.5%

         \[\leadsto e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(\frac{\phi_1 + \phi_2}{2}\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right)\right)} - 1 \]

      4. div-inv 43.5%

         \[\leadsto e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)} \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1 \]

      5. metadata-eval 43.5%

         \[\leadsto e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1 \]

   6. Applied egg-rr 43.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]

   8. Simplified 66.1%

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

   9. Taylor expanded in phi2 around inf 32.0%

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

       1. mul-1-neg 32.0%

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

       2. unsub-neg 32.0%

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

       3. associate-/l* 31.3%

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

   11. Simplified 31.3%

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

   if 1.85e-50 < lambda1

   1. Initial program 48.9%

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

      1. hypot-define 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in lambda2 around inf 14.9%

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

      1. *-commutative 14.9%

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

      2. associate-*r* 14.9%

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

      3. *-commutative 14.9%

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

      4. associate-*l* 14.9%

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

      5. +-commutative 14.9%

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

   7. Simplified 14.9%

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

   8. Taylor expanded in phi1 around 0 13.0%

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

      1. associate-*r* 13.0%

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

   10. Simplified 13.0%

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

   11. Taylor expanded in phi2 around 0 14.3%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.1 \cdot 10^{+158}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 30.7% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.9 \cdot 10^{+159}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -1.9e+159)
   (* R (- lambda1))
   (if (<= lambda1 1.85e-50)
     (* phi2 (- R (/ (* R phi1) phi2)))
     (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -1.9e+159) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 1.85e-50) {
		tmp = phi2 * (R - ((R * phi1) / 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 (lambda1 <= (-1.9d+159)) then
        tmp = r * -lambda1
    else if (lambda1 <= 1.85d-50) then
        tmp = phi2 * (r - ((r * phi1) / 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 (lambda1 <= -1.9e+159) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 1.85e-50) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -1.9e+159:
		tmp = R * -lambda1
	elif lambda1 <= 1.85e-50:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -1.9e+159)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda1 <= 1.85e-50)
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -1.9e+159)
		tmp = R * -lambda1;
	elseif (lambda1 <= 1.85e-50)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -1.9e+159], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, 1.85e-50], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.89999999999999983e159

   1. Initial program 40.7%

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

      1. hypot-define 91.9%

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

   3. Simplified 91.9%

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

   5. Taylor expanded in phi1 around 0 40.7%

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

      1. +-commutative 40.7%

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

      2. unpow2 40.7%

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

      3. unpow2 40.7%

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

      4. unpow2 40.7%

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

      5. unswap-sqr 40.7%

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

      6. hypot-define 73.3%

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

   7. Simplified 73.3%

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

   8. Taylor expanded in lambda2 around inf 45.3%

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

      1. +-commutative 45.3%

         \[\leadsto \lambda_2 \cdot \color{blue}{\left(R \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}{\lambda_2}\right)} \]

      2. mul-1-neg 45.3%

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

      3. unsub-neg 45.3%

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

      4. *-commutative 45.3%

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

      5. associate-/l* 35.1%

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

      6. *-commutative 35.1%

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

   10. Simplified 35.1%

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

   11. Taylor expanded in lambda2 around 0 45.7%

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

       1. *-commutative 45.7%

          \[\leadsto -1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)\right) \]

       2. *-commutative 45.7%

          \[\leadsto -1 \cdot \left(R \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_1\right)}\right) \]

       3. neg-mul-1 45.7%

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

       4. distribute-rgt-neg-in 45.7%

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

       5. *-commutative 45.7%

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

       6. distribute-rgt-neg-in 45.7%

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

   13. Simplified 45.7%

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

   14. Taylor expanded in phi2 around 0 57.5%

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

       1. mul-1-neg 57.5%

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

       2. distribute-rgt-neg-in 57.5%

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

   16. Simplified 57.5%

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

   if -1.89999999999999983e159 < lambda1 < 1.85e-50

   1. Initial program 63.0%

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

      1. hypot-define 97.9%

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

   3. Simplified 97.9%

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

   5. Taylor expanded in phi2 around inf 32.0%

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

      1. associate-*r/ 32.0%

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

      2. mul-1-neg 32.0%

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

      3. *-commutative 32.0%

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

   7. Simplified 32.0%

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

   if 1.85e-50 < lambda1

   1. Initial program 48.9%

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

      1. hypot-define 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in lambda2 around inf 14.9%

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

      1. *-commutative 14.9%

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

      2. associate-*r* 14.9%

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

      3. *-commutative 14.9%

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

      4. associate-*l* 14.9%

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

      5. +-commutative 14.9%

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

   7. Simplified 14.9%

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

   8. Taylor expanded in phi1 around 0 13.0%

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

      1. associate-*r* 13.0%

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

   10. Simplified 13.0%

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

   11. Taylor expanded in phi2 around 0 14.3%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.9 \cdot 10^{+159}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 1.85 \cdot 10^{-50}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 24.5% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.8 \cdot 10^{+37}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 6.4 \cdot 10^{-135}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4.8e+37)
   (* lambda1 (- R))
   (if (<= lambda1 6.4e-135) (* R phi2) (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -4.8e+37) {
		tmp = lambda1 * -R;
	} else if (lambda1 <= 6.4e-135) {
		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 (lambda1 <= (-4.8d+37)) then
        tmp = lambda1 * -r
    else if (lambda1 <= 6.4d-135) 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 (lambda1 <= -4.8e+37) {
		tmp = lambda1 * -R;
	} else if (lambda1 <= 6.4e-135) {
		tmp = R * phi2;
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -4.8e+37:
		tmp = lambda1 * -R
	elif lambda1 <= 6.4e-135:
		tmp = R * phi2
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -4.8e+37)
		tmp = Float64(lambda1 * Float64(-R));
	elseif (lambda1 <= 6.4e-135)
		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 (lambda1 <= -4.8e+37)
		tmp = lambda1 * -R;
	elseif (lambda1 <= 6.4e-135)
		tmp = R * phi2;
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -4.8e+37], N[(lambda1 * (-R)), $MachinePrecision], If[LessEqual[lambda1, 6.4e-135], N[(R * phi2), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -4.8e37

   1. Initial program 49.6%

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

      1. hypot-define 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in phi1 around 0 43.9%

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

      1. +-commutative 43.9%

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

      2. unpow2 43.9%

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

      3. unpow2 43.9%

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

      4. unpow2 43.9%

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

      5. unswap-sqr 43.9%

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

      6. hypot-define 67.1%

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

   7. Simplified 67.1%

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

   8. Taylor expanded in lambda2 around inf 43.7%

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

      1. +-commutative 43.7%

         \[\leadsto \lambda_2 \cdot \color{blue}{\left(R \cdot \cos \left(0.5 \cdot \phi_2\right) + -1 \cdot \frac{R \cdot \left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)}{\lambda_2}\right)} \]

      2. mul-1-neg 43.7%

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

      3. unsub-neg 43.7%

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

      4. *-commutative 43.7%

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

      5. associate-/l* 35.8%

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

      6. *-commutative 35.8%

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

   10. Simplified 35.8%

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

   11. Taylor expanded in lambda2 around 0 42.4%

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

       1. *-commutative 42.4%

          \[\leadsto -1 \cdot \left(R \cdot \left(\lambda_1 \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right)\right) \]

       2. *-commutative 42.4%

          \[\leadsto -1 \cdot \left(R \cdot \color{blue}{\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_1\right)}\right) \]

       3. neg-mul-1 42.4%

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

       4. distribute-rgt-neg-in 42.4%

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

       5. *-commutative 42.4%

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

       6. distribute-rgt-neg-in 42.4%

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

   13. Simplified 42.4%

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

   14. Taylor expanded in phi2 around 0 48.5%

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

       1. mul-1-neg 48.5%

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

       2. distribute-rgt-neg-in 48.5%

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

   16. Simplified 48.5%

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

   if -4.8e37 < lambda1 < 6.40000000000000001e-135

   1. Initial program 63.7%

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

      1. hypot-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in phi2 around inf 21.9%

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

      1. *-commutative 21.9%

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

   7. Simplified 21.9%

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

   if 6.40000000000000001e-135 < lambda1

   1. Initial program 49.9%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in lambda2 around inf 13.5%

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

      1. *-commutative 13.5%

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

      2. associate-*r* 13.5%

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

      3. *-commutative 13.5%

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

      4. associate-*l* 13.5%

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

      5. +-commutative 13.5%

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

   7. Simplified 13.5%

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

   8. Taylor expanded in phi1 around 0 12.9%

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

      1. associate-*r* 12.9%

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

   10. Simplified 12.9%

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

   11. Taylor expanded in phi2 around 0 14.3%

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

4. Final simplification 24.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.8 \cdot 10^{+37}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\lambda_1 \leq 6.4 \cdot 10^{-135}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 28.8% accurate, 25.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.56 \cdot 10^{-139}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.8e-25)
   (* R (- phi1))
   (if (<= phi1 -1.56e-139) (* R lambda2) (* R phi2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.8e-25) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.56e-139) {
		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 (phi1 <= (-6.8d-25)) then
        tmp = r * -phi1
    else if (phi1 <= (-1.56d-139)) 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 (phi1 <= -6.8e-25) {
		tmp = R * -phi1;
	} else if (phi1 <= -1.56e-139) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.8e-25:
		tmp = R * -phi1
	elif phi1 <= -1.56e-139:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.8e-25)
		tmp = Float64(R * Float64(-phi1));
	elseif (phi1 <= -1.56e-139)
		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 (phi1 <= -6.8e-25)
		tmp = R * -phi1;
	elseif (phi1 <= -1.56e-139)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.8e-25], N[(R * (-phi1)), $MachinePrecision], If[LessEqual[phi1, -1.56e-139], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -6.80000000000000003e-25

   1. Initial program 53.2%

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

      1. hypot-define 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in phi1 around -inf 51.2%

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

      1. mul-1-neg 51.2%

         \[\leadsto \color{blue}{-R \cdot \phi_1} \]

      2. *-commutative 51.2%

         \[\leadsto -\color{blue}{\phi_1 \cdot R} \]

      3. distribute-rgt-neg-in 51.2%

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

   7. Simplified 51.2%

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

   if -6.80000000000000003e-25 < phi1 < -1.56000000000000008e-139

   1. Initial program 39.0%

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

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in lambda2 around inf 22.6%

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

      1. *-commutative 22.6%

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

      2. associate-*r* 22.6%

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

      3. *-commutative 22.6%

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

      4. associate-*l* 22.6%

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

      5. +-commutative 22.6%

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

   7. Simplified 22.6%

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

   8. Taylor expanded in phi1 around 0 22.6%

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

      1. associate-*r* 22.6%

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

   10. Simplified 22.6%

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

   11. Taylor expanded in phi2 around 0 20.3%

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

   if -1.56000000000000008e-139 < phi1

   1. Initial program 60.1%

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

      1. hypot-define 98.2%

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

   3. Simplified 98.2%

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

   5. Taylor expanded in phi2 around inf 19.5%

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

      1. *-commutative 19.5%

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

   7. Simplified 19.5%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.8 \cdot 10^{-25}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq -1.56 \cdot 10^{-139}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.1% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3100000000000:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3100000000000.0) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3100000000000.0) {
		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 <= 3100000000000.0d0) 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 <= 3100000000000.0) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3100000000000.0:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3100000000000.0)
		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 <= 3100000000000.0)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3100000000000.0], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.1e12

   1. Initial program 57.1%

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

      1. hypot-define 97.0%

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

   3. Simplified 97.0%

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

   5. Taylor expanded in lambda2 around inf 14.1%

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

      1. *-commutative 14.1%

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

      2. associate-*r* 14.1%

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

      3. *-commutative 14.1%

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

      4. associate-*l* 14.1%

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

      5. +-commutative 14.1%

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

   7. Simplified 14.1%

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

   8. Taylor expanded in phi1 around 0 11.2%

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

      1. associate-*r* 11.2%

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

   10. Simplified 11.2%

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

   11. Taylor expanded in phi2 around 0 12.1%

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

   if 3.1e12 < phi2

   1. Initial program 51.3%

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

      1. hypot-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in phi2 around inf 55.4%

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

      1. *-commutative 55.4%

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

   7. Simplified 55.4%

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

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3100000000000:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 14.0% accurate, 109.7× speedup

Math

\[\begin{array}{l} \\ R \cdot \lambda_2 \end{array} \]

FPCore

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

C

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

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 * lambda2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * lambda2), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

   1. hypot-define 96.9%

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

3. Simplified 96.9%

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

5. Taylor expanded in lambda2 around inf 14.8%

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

   1. *-commutative 14.8%

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

   2. associate-*r* 14.7%

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

   3. *-commutative 14.7%

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

   4. associate-*l* 14.7%

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

   5. +-commutative 14.7%

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

7. Simplified 14.7%

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

8. Taylor expanded in phi1 around 0 12.0%

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

   1. associate-*r* 12.0%

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

10. Simplified 12.0%

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

11. Taylor expanded in phi2 around 0 11.5%

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

12. Final simplification 11.5%

    \[\leadsto R \cdot \lambda_2 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(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))))))