Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.9% → 98.6%

Time: 21.1s

Alternatives: 16

Speedup: 3.0×

• 31.6% 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: 59.9% 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: 98.6% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. hypot-define 95.9%

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

3. Simplified 95.9%

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

   1. add-cube-cbrt 95.3%

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

   2. pow3 95.3%

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

6. Applied egg-rr 95.3%

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

   1. add-cube-cbrt 94.5%

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

   2. pow3 94.5%

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

8. Applied egg-rr 94.6%

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

   1. distribute-lft-in 94.6%

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

   2. cos-sum 98.6%

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

10. Applied egg-rr 98.6%

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

    1. fmm-def 98.6%

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

12. Simplified 98.6%

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

13. Final simplification 98.6%

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

Alternative 2: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

1. Initial program 60.3%

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

   1. hypot-define 95.9%

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

3. Simplified 95.9%

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

   1. add-cube-cbrt 95.3%

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

   2. pow3 95.3%

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

6. Applied egg-rr 95.3%

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

   1. add-cube-cbrt 94.5%

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

   2. pow3 94.5%

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

8. Applied egg-rr 94.6%

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

   1. distribute-lft-in 94.6%

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

   2. cos-sum 98.6%

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

10. Applied egg-rr 98.6%

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

Alternative 3: 93.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -18:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -18.0)
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi1 2.0))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi2 2.0))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -18.0) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -18.0) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -18.0:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -18.0)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -18.0)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -18.0], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -18

   1. Initial program 49.2%

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

      1. hypot-define 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in phi1 around inf 87.3%

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

   if -18 < phi1

   1. Initial program 64.7%

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

      1. hypot-define 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in phi1 around 0 95.2%

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

Alternative 4: 91.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.033:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 0.033)
   (* R (hypot (* (- lambda1 lambda2) (cos (/ phi1 2.0))) (- phi1 phi2)))
   (* R (hypot (* lambda2 (- (cos (* 0.5 phi2)))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.033) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * -cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 0.033) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * -Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.033:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi1 / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * -math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.033)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(-cos(Float64(0.5 * phi2)))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 0.033)
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi1 / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * -cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 0.033], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * (-N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.033000000000000002

   1. Initial program 63.1%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi1 around inf 92.2%

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

   if 0.033000000000000002 < phi2

   1. Initial program 51.6%

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

      1. hypot-define 96.1%

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

   3. Simplified 96.1%

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

      1. expm1-log1p-u 75.6%

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

      2. expm1-undefine 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. expm1-define 75.6%

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

      2. +-commutative 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in phi1 around 0 89.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 89.0%

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

   11. Simplified 89.0%

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

   12. Taylor expanded in lambda1 around 0 92.7%

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

       1. mul-1-neg 92.7%

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

       2. distribute-lft-neg-out 92.7%

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

   14. Simplified 92.7%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.033:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(-\cos \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.0% accurate, 1.5× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e-5)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot (* lambda2 (- (cos (* 0.5 phi2)))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-5) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot((lambda2 * -cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-5) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot((lambda2 * -Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e-5:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot((lambda2 * -math.cos((0.5 * phi2))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e-5)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(-cos(Float64(0.5 * phi2)))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5e-5)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot((lambda2 * -cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e-5], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * (-N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.00000000000000024e-5

   1. Initial program 63.1%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

      1. expm1-log1p-u 48.7%

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

      2. expm1-undefine 48.7%

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

   6. Applied egg-rr 48.7%

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

      1. expm1-define 48.7%

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

      2. +-commutative 48.7%

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

   8. Simplified 48.7%

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

   9. Taylor expanded in phi1 around 0 58.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 58.8%

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

   11. Simplified 58.8%

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

   12. Taylor expanded in phi2 around 0 54.2%

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

       1. unpow2 54.2%

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

       2. unpow2 54.2%

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

       3. hypot-define 72.8%

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

   14. Simplified 72.8%

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

   if 5.00000000000000024e-5 < phi2

   1. Initial program 51.6%

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

      1. hypot-define 96.1%

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

   3. Simplified 96.1%

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

      1. expm1-log1p-u 75.6%

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

      2. expm1-undefine 75.6%

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

   6. Applied egg-rr 75.6%

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

      1. expm1-define 75.6%

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

      2. +-commutative 75.6%

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

   8. Simplified 75.6%

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

   9. Taylor expanded in phi1 around 0 89.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 89.0%

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

   11. Simplified 89.0%

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

   12. Taylor expanded in lambda1 around 0 92.7%

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

       1. mul-1-neg 92.7%

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

       2. distribute-lft-neg-out 92.7%

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

   14. Simplified 92.7%

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

4. Final simplification 77.5%

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

Alternative 6: 95.9% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 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[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 60.3%

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

   1. hypot-define 95.9%

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

3. Simplified 95.9%

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

Alternative 7: 69.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1100000000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1100000000000.0)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1100000000000.0) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1100000000000.0) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1100000000000.0:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1100000000000.0)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1100000000000.0)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1100000000000.0], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.1e12

   1. Initial program 63.4%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

      1. expm1-log1p-u 49.3%

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

      2. expm1-undefine 49.2%

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

   6. Applied egg-rr 49.2%

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

      1. expm1-define 49.3%

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

      2. +-commutative 49.3%

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

   8. Simplified 49.3%

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

   9. Taylor expanded in phi1 around 0 59.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 59.2%

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

   11. Simplified 59.2%

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

   12. Taylor expanded in phi2 around 0 54.2%

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

       1. unpow2 54.2%

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

       2. unpow2 54.2%

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

       3. hypot-define 72.6%

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

   14. Simplified 72.6%

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

   if 1.1e12 < phi2

   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 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in phi2 around inf 68.6%

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

      1. mul-1-neg 68.6%

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

      2. unsub-neg 68.6%

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

   7. Simplified 68.6%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 30.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.7 \cdot 10^{+126}:\\ \;\;\;\;\left|R \cdot \lambda_1\right|\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -8.7e+126)
   (fabs (* R lambda1))
   (* phi2 (- R (/ (* R phi1) phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -8.7e+126) {
		tmp = fabs((R * lambda1));
	} else {
		tmp = phi2 * (R - ((R * phi1) / 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 (lambda1 <= (-8.7d+126)) then
        tmp = abs((r * lambda1))
    else
        tmp = phi2 * (r - ((r * phi1) / 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 (lambda1 <= -8.7e+126) {
		tmp = Math.abs((R * lambda1));
	} else {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -8.7e+126:
		tmp = math.fabs((R * lambda1))
	else:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -8.7e+126)
		tmp = abs(Float64(R * lambda1));
	else
		tmp = Float64(phi2 * Float64(R - Float64(Float64(R * phi1) / phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -8.7e+126)
		tmp = abs((R * lambda1));
	else
		tmp = phi2 * (R - ((R * phi1) / phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -8.7e+126], N[Abs[N[(R * lambda1), $MachinePrecision]], $MachinePrecision], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -8.7000000000000002e126

   1. Initial program 48.6%

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

      1. hypot-define 86.2%

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

   3. Simplified 86.2%

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

   5. Taylor expanded in lambda1 around inf 25.7%

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

      1. *-commutative 25.7%

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

   7. Simplified 25.7%

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

   8. Taylor expanded in phi2 around 0 24.1%

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

   9. Taylor expanded in phi1 around 0 0.6%

      \[\leadsto \color{blue}{\lambda_1} \cdot R \]
   10. Step-by-step derivation

       1. add-sqr-sqrt 0.4%

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

       2. sqrt-unprod 25.9%

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

       3. pow2 25.9%

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

       4. *-commutative 25.9%

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

   11. Applied egg-rr 25.9%

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

       1. unpow2 25.9%

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

       2. rem-sqrt-square 26.4%

          \[\leadsto \color{blue}{\left|R \cdot \lambda_1\right|} \]

   13. Simplified 26.4%

       \[\leadsto \color{blue}{\left|R \cdot \lambda_1\right|} \]

   if -8.7000000000000002e126 < lambda1

   1. Initial program 62.7%

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

      1. hypot-define 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 33.2%

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

      1. associate-*r/ 33.2%

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

      2. mul-1-neg 33.2%

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

      3. *-commutative 33.2%

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

   7. Simplified 33.2%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -8.7 \cdot 10^{+126}:\\ \;\;\;\;\left|R \cdot \lambda_1\right|\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.5 \cdot 10^{+109}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.5e+109) (* phi2 (- R (/ (* R phi1) phi2))) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.5e+109) {
		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 (lambda2 <= 1.5d+109) 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 (lambda2 <= 1.5e+109) {
		tmp = phi2 * (R - ((R * phi1) / phi2));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.5e+109:
		tmp = phi2 * (R - ((R * phi1) / phi2))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.5e+109)
		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 (lambda2 <= 1.5e+109)
		tmp = phi2 * (R - ((R * phi1) / phi2));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.5e+109], N[(phi2 * N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.50000000000000008e109

   1. Initial program 62.5%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi2 around inf 33.2%

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

      1. associate-*r/ 33.2%

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

      2. mul-1-neg 33.2%

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

      3. *-commutative 33.2%

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

   7. Simplified 33.2%

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

   if 1.50000000000000008e109 < lambda2

   1. Initial program 44.8%

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

      1. hypot-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. Step-by-step derivation

      1. expm1-log1p-u 47.3%

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

      2. expm1-undefine 47.3%

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

   6. Applied egg-rr 47.3%

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

      1. expm1-define 47.3%

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

      2. +-commutative 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in phi1 around 0 46.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 46.3%

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

   11. Simplified 46.3%

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

   12. Taylor expanded in lambda2 around inf 43.3%

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

       1. associate-*r* 43.3%

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

   14. Simplified 43.3%

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

   15. Taylor expanded in phi2 around 0 54.2%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.5 \cdot 10^{+109}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{R \cdot \phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.5% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.8 \cdot 10^{+109}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.8e+109) (* phi1 (- (* R (/ phi2 phi1)) R)) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.8e+109) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 1.8d+109) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.8e+109) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.8e+109:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.8e+109)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.8e+109)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.8e+109], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.8e109

   1. Initial program 62.5%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi1 around -inf 31.1%

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

      1. mul-1-neg 31.1%

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

      2. distribute-rgt-neg-in 31.1%

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

      3. mul-1-neg 31.1%

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

      4. unsub-neg 31.1%

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

      5. associate-/l* 29.7%

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

   7. Simplified 29.7%

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

   if 1.8e109 < lambda2

   1. Initial program 44.8%

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

      1. hypot-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. Step-by-step derivation

      1. expm1-log1p-u 47.3%

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

      2. expm1-undefine 47.3%

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

   6. Applied egg-rr 47.3%

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

      1. expm1-define 47.3%

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

      2. +-commutative 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in phi1 around 0 46.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 46.3%

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

   11. Simplified 46.3%

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

   12. Taylor expanded in lambda2 around inf 43.3%

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

       1. associate-*r* 43.3%

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

   14. Simplified 43.3%

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

   15. Taylor expanded in phi2 around 0 54.2%

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

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.8 \cdot 10^{+109}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.4 \cdot 10^{+108}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 4.4e+108)
   (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
   (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4.4e+108) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 4.4d+108) then
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4.4e+108) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 4.4e+108:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 4.4e+108)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 4.4e+108)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 4.4e+108], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 4.4000000000000003e108

   1. Initial program 62.5%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi1 around -inf 29.3%

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

      1. mul-1-neg 29.3%

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

      2. distribute-rgt-neg-in 29.3%

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

      3. mul-1-neg 29.3%

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

      4. unsub-neg 29.3%

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

   7. Simplified 29.3%

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

   if 4.4000000000000003e108 < lambda2

   1. Initial program 44.8%

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

      1. hypot-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. Step-by-step derivation

      1. expm1-log1p-u 47.3%

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

      2. expm1-undefine 47.3%

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

   6. Applied egg-rr 47.3%

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

      1. expm1-define 47.3%

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

      2. +-commutative 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in phi1 around 0 46.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 46.3%

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

   11. Simplified 46.3%

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

   12. Taylor expanded in lambda2 around inf 43.3%

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

       1. associate-*r* 43.3%

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

   14. Simplified 43.3%

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

   15. Taylor expanded in phi2 around 0 54.2%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4.4 \cdot 10^{+108}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 33.7% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.55 \cdot 10^{+109}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.55e+109) (* phi2 (- R (* R (/ phi1 phi2)))) (* R lambda2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.55e+109) {
		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 (lambda2 <= 1.55d+109) 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 (lambda2 <= 1.55e+109) {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.55e+109:
		tmp = phi2 * (R - (R * (phi1 / phi2)))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.55e+109)
		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 (lambda2 <= 1.55e+109)
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.55e+109], N[(phi2 * N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.54999999999999996e109

   1. Initial program 62.5%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi2 around inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. unsub-neg 33.2%

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

      3. associate-/l* 31.0%

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

   7. Simplified 31.0%

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

   if 1.54999999999999996e109 < lambda2

   1. Initial program 44.8%

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

      1. hypot-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. Step-by-step derivation

      1. expm1-log1p-u 47.3%

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

      2. expm1-undefine 47.3%

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

   6. Applied egg-rr 47.3%

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

      1. expm1-define 47.3%

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

      2. +-commutative 47.3%

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

   8. Simplified 47.3%

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

   9. Taylor expanded in phi1 around 0 46.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 46.3%

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

   11. Simplified 46.3%

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

   12. Taylor expanded in lambda2 around inf 43.3%

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

       1. associate-*r* 43.3%

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

   14. Simplified 43.3%

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

   15. Taylor expanded in phi2 around 0 54.2%

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

Alternative 13: 30.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 4.8 \cdot 10^{-123}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 4.8e-123) (* R (- phi1)) (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.8e-123) {
		tmp = R * -phi1;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 4.8d-123) then
        tmp = r * -phi1
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 4.8e-123) {
		tmp = R * -phi1;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 4.8e-123:
		tmp = R * -phi1
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 4.8e-123)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 4.8e-123)
		tmp = R * -phi1;
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 4.8e-123], N[(R * (-phi1)), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 4.8e-123

   1. Initial program 59.8%

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

      1. hypot-define 95.1%

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

   3. Simplified 95.1%

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

   5. Taylor expanded in phi1 around -inf 17.6%

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

      1. mul-1-neg 17.6%

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

   7. Simplified 17.6%

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

   if 4.8e-123 < phi2

   1. Initial program 61.2%

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

      1. hypot-define 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 52.9%

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

      1. mul-1-neg 52.9%

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

      2. unsub-neg 52.9%

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

   7. Simplified 52.9%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 29.2% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.2e-27) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.2e-27) {
		tmp = R * -phi1;
	} 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 <= (-3.2d-27)) then
        tmp = r * -phi1
    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 <= -3.2e-27) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.2e-27:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.2e-27)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.2e-27)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.2e-27], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.19999999999999991e-27

   1. Initial program 50.6%

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

      1. hypot-define 87.5%

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

   3. Simplified 87.5%

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

   5. Taylor expanded in phi1 around -inf 51.9%

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

      1. mul-1-neg 51.9%

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

   7. Simplified 51.9%

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

   if -3.19999999999999991e-27 < phi1

   1. Initial program 64.4%

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

      1. hypot-define 99.3%

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

   3. Simplified 99.3%

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

   5. Taylor expanded in phi2 around inf 19.4%

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

      1. *-commutative 19.4%

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

   7. Simplified 19.4%

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

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.2 \cdot 10^{-27}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 26.2% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e-36) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-36) {
		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 <= 5d-36) 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 <= 5e-36) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e-36:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e-36)
		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 <= 5e-36)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e-36], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.00000000000000004e-36

   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 95.7%

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

   3. Simplified 95.7%

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

      1. expm1-log1p-u 48.5%

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

      2. expm1-undefine 48.4%

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

   6. Applied egg-rr 48.4%

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

      1. expm1-define 48.5%

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

      2. +-commutative 48.5%

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

   8. Simplified 48.5%

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

   9. Taylor expanded in phi1 around 0 58.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
   10. Step-by-step derivation

       1. log1p-define 58.2%

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

   11. Simplified 58.2%

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

   12. Taylor expanded in lambda2 around inf 10.2%

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

       1. associate-*r* 10.2%

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

   14. Simplified 10.2%

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

   15. Taylor expanded in phi2 around 0 11.6%

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

   if 5.00000000000000004e-36 < phi2

   1. Initial program 52.4%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi2 around inf 60.4%

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

      1. *-commutative 60.4%

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

   7. Simplified 60.4%

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

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-36}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.7% 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 60.3%

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

   1. hypot-define 95.9%

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

3. Simplified 95.9%

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

   1. expm1-log1p-u 55.1%

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

   2. expm1-undefine 55.1%

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

6. Applied egg-rr 55.1%

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

   1. expm1-define 55.1%

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

   2. +-commutative 55.1%

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

8. Simplified 55.1%

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

9. Taylor expanded in phi1 around 0 66.0%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\mathsf{expm1}\left(\color{blue}{\log \left(1 + \phi_2\right)}\right)}{2}\right), \phi_1 - \phi_2\right) \]
10. Step-by-step derivation

    1. log1p-define 66.0%

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

11. Simplified 66.0%

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

12. Taylor expanded in lambda2 around inf 10.6%

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

    1. associate-*r* 10.6%

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

14. Simplified 10.6%

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

15. Taylor expanded in phi2 around 0 10.7%

    \[\leadsto \color{blue}{R \cdot \lambda_2} \]
16. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(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))))))