Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 99.9%

Time: 21.9s

Alternatives: 9

Speedup: 1.6×

• 30.8% 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: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(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]

Derivation

1. Initial program 60.1%

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

   1. hypot-def 96.6%

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

3. Simplified 96.6%

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

   1. add-cube-cbrt 95.4%

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

   2. pow3 95.3%

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

   3. *-commutative 95.3%

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

   4. div-inv 95.3%

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

   5. metadata-eval 95.3%

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

5. Applied egg-rr 95.3%

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

   1. *-commutative 95.3%

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

   2. +-commutative 95.3%

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

   3. distribute-rgt-in 95.3%

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

   4. cos-sum 98.5%

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

7. Applied egg-rr 98.5%

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

   1. fma-neg 98.5%

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

   2. *-commutative 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. *-commutative 98.5%

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

   6. distribute-rgt-neg-in 98.5%

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

9. Simplified 98.5%

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

    1. rem-cube-cbrt 99.8%

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

    2. expm1-log1p-u 54.0%

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

    3. expm1-udef 39.2%

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

11. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 54.0%

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

    2. expm1-log1p 99.8%

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

    3. *-commutative 99.8%

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

    4. distribute-rgt-neg-in 99.8%

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

    5. fma-neg 99.8%

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

    6. *-commutative 99.8%

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

    7. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 96.1% accurate, 1.5× speedup

Math

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

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 60.1%

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

   1. hypot-def 96.6%

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

3. Simplified 96.6%

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

4. Final simplification 96.6%

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

Alternative 3: 76.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e+51)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (- phi2 phi1))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e+51) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e+51) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e+51:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * (phi2 - phi1)
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e+51)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5e+51)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * (phi2 - phi1);
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e+51], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5e51

   1. Initial program 63.3%

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

      1. hypot-def 97.7%

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

   3. Simplified 97.7%

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

   4. Taylor expanded in phi2 around 0 55.0%

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

      1. +-commutative 55.0%

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

      2. unpow2 55.0%

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

      3. unpow2 55.0%

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

      4. unpow2 55.0%

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

      5. unswap-sqr 55.0%

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

      6. hypot-def 78.0%

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

   6. Simplified 78.0%

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

   if 5e51 < phi2

   1. Initial program 46.0%

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

      1. hypot-def 91.9%

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

   3. Simplified 91.9%

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

      1. add-cube-cbrt 90.8%

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

      2. pow3 90.8%

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

      3. *-commutative 90.8%

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

      4. div-inv 90.8%

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

      5. metadata-eval 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. *-commutative 90.8%

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

      2. +-commutative 90.8%

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

      3. distribute-rgt-in 90.8%

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

      4. cos-sum 98.6%

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

   7. Applied egg-rr 98.6%

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

      1. fma-neg 98.6%

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

      2. *-commutative 98.6%

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

      3. *-commutative 98.6%

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

      4. *-commutative 98.6%

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

      5. *-commutative 98.6%

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

      6. distribute-rgt-neg-in 98.6%

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

   9. Simplified 98.6%

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

       1. rem-cube-cbrt 99.8%

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

       2. expm1-log1p-u 50.3%

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

       3. expm1-udef 38.1%

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

   11. Applied egg-rr 38.1%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 50.3%

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

       2. expm1-log1p 99.8%

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

       3. *-commutative 99.8%

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

       4. distribute-rgt-neg-in 99.8%

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

       5. fma-neg 99.9%

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

       6. *-commutative 99.9%

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

       7. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   14. Taylor expanded in phi1 around -inf 65.6%

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

       1. +-commutative 65.6%

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

       2. mul-1-neg 65.6%

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

       3. sub-neg 65.6%

          \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

       4. *-commutative 65.6%

          \[\leadsto R \cdot \phi_2 - \color{blue}{\phi_1 \cdot R} \]

       5. cancel-sign-sub-inv 65.6%

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

       6. *-commutative 65.6%

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

       7. neg-mul-1 65.6%

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

       8. distribute-rgt-out 67.7%

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

       9. neg-mul-1 67.7%

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

       10. sub-neg 67.7%

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

   16. Simplified 67.7%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]

Alternative 4: 79.6% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0255:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.0255)
   (* R (hypot phi1 (* (- lambda1 lambda2) (cos (* 0.5 phi1)))))
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* 0.5 phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0255) {
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.0255) {
		tmp = R * Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((0.5 * phi1))));
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((0.5 * phi2))));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.0255:
		tmp = R * math.hypot(phi1, ((lambda1 - lambda2) * math.cos((0.5 * phi1))))
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((0.5 * phi2))))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.0255)
		tmp = Float64(R * hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -0.0255)
		tmp = R * hypot(phi1, ((lambda1 - lambda2) * cos((0.5 * phi1))));
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((0.5 * phi2))));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.0255], N[(R * N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.0254999999999999984

   1. Initial program 48.2%

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

      1. hypot-def 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. Taylor expanded in phi2 around 0 44.3%

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

      1. +-commutative 44.3%

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

      2. unpow2 44.3%

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

      3. unpow2 44.3%

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

      4. unpow2 44.3%

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

      5. unswap-sqr 44.3%

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

      6. hypot-def 79.8%

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

   6. Simplified 79.8%

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

   if -0.0254999999999999984 < phi1

   1. Initial program 62.9%

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

      1. hypot-def 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. Taylor expanded in phi1 around 0 55.5%

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

      1. +-commutative 55.5%

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

      2. unpow2 55.5%

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

      3. unpow2 55.5%

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

      4. unpow2 55.5%

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

      5. unswap-sqr 55.5%

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

      6. hypot-def 80.5%

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

   6. Simplified 80.5%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.0255:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 5: 33.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.15 \cdot 10^{+172}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.15e+172)
   (* R (- phi2 phi1))
   (* R (* lambda2 (cos (* 0.5 phi1))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.15e+172) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * cos((0.5 * phi1)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 1.15d+172) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (lambda2 * cos((0.5d0 * phi1)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.15e+172) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * Math.cos((0.5 * phi1)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.15e+172:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (lambda2 * math.cos((0.5 * phi1)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.15e+172)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(lambda2 * cos(Float64(0.5 * phi1))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.15e+172)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (lambda2 * cos((0.5 * phi1)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.15e+172], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.15e172

   1. Initial program 62.4%

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

      1. hypot-def 96.9%

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

   3. Simplified 96.9%

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

      1. add-cube-cbrt 95.6%

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

      2. pow3 95.5%

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

      3. *-commutative 95.5%

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

      4. div-inv 95.5%

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

      5. metadata-eval 95.5%

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

   5. Applied egg-rr 95.5%

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

      1. *-commutative 95.5%

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

      2. +-commutative 95.5%

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

      3. distribute-rgt-in 95.5%

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

      4. cos-sum 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. fma-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

      5. *-commutative 98.5%

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

      6. distribute-rgt-neg-in 98.5%

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

   9. Simplified 98.5%

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

       1. rem-cube-cbrt 99.8%

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

       2. expm1-log1p-u 55.4%

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

       3. expm1-udef 39.8%

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

   11. Applied egg-rr 39.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 55.4%

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

       2. expm1-log1p 99.8%

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

       3. *-commutative 99.8%

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

       4. distribute-rgt-neg-in 99.8%

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

       5. fma-neg 99.9%

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

       6. *-commutative 99.9%

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

       7. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   14. Taylor expanded in phi1 around -inf 25.6%

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

       1. +-commutative 25.6%

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

       2. mul-1-neg 25.6%

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

       3. sub-neg 25.6%

          \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

       4. *-commutative 25.6%

          \[\leadsto R \cdot \phi_2 - \color{blue}{\phi_1 \cdot R} \]

       5. cancel-sign-sub-inv 25.6%

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

       6. *-commutative 25.6%

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

       7. neg-mul-1 25.6%

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

       8. distribute-rgt-out 26.9%

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

       9. neg-mul-1 26.9%

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

       10. sub-neg 26.9%

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

   16. Simplified 26.9%

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

   if 1.15e172 < lambda2

   1. Initial program 45.1%

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

      1. hypot-def 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in lambda2 around inf 55.2%

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

      1. *-commutative 55.2%

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

      2. *-commutative 55.2%

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

      3. *-commutative 55.2%

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

      4. associate-*l* 55.2%

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

      5. *-commutative 55.2%

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

      6. +-commutative 55.2%

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

   6. Simplified 55.2%

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

   7. Taylor expanded in phi2 around 0 59.7%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.15 \cdot 10^{+172}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_1\right)\right)\\ \end{array} \]

Alternative 6: 33.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.25 \cdot 10^{+172}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.25e+172)
   (* R (- phi2 phi1))
   (* R (* lambda2 (cos (* 0.5 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.25e+172) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * cos((0.5 * 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 (lambda2 <= 1.25d+172) then
        tmp = r * (phi2 - phi1)
    else
        tmp = r * (lambda2 * cos((0.5d0 * 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 (lambda2 <= 1.25e+172) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = R * (lambda2 * Math.cos((0.5 * phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.25e+172:
		tmp = R * (phi2 - phi1)
	else:
		tmp = R * (lambda2 * math.cos((0.5 * phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.25e+172)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(R * Float64(lambda2 * cos(Float64(0.5 * phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.25e+172)
		tmp = R * (phi2 - phi1);
	else
		tmp = R * (lambda2 * cos((0.5 * phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.25e+172], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(R * N[(lambda2 * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.25e172

   1. Initial program 62.4%

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

      1. hypot-def 96.9%

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

   3. Simplified 96.9%

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

      1. add-cube-cbrt 95.6%

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

      2. pow3 95.5%

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

      3. *-commutative 95.5%

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

      4. div-inv 95.5%

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

      5. metadata-eval 95.5%

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

   5. Applied egg-rr 95.5%

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

      1. *-commutative 95.5%

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

      2. +-commutative 95.5%

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

      3. distribute-rgt-in 95.5%

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

      4. cos-sum 98.5%

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

   7. Applied egg-rr 98.5%

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

      1. fma-neg 98.5%

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

      2. *-commutative 98.5%

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

      3. *-commutative 98.5%

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

      4. *-commutative 98.5%

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

      5. *-commutative 98.5%

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

      6. distribute-rgt-neg-in 98.5%

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

   9. Simplified 98.5%

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

       1. rem-cube-cbrt 99.8%

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

       2. expm1-log1p-u 55.4%

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

       3. expm1-udef 39.8%

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

   11. Applied egg-rr 39.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 55.4%

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

       2. expm1-log1p 99.8%

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

       3. *-commutative 99.8%

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

       4. distribute-rgt-neg-in 99.8%

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

       5. fma-neg 99.9%

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

       6. *-commutative 99.9%

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

       7. *-commutative 99.9%

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

   13. Simplified 99.9%

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

   14. Taylor expanded in phi1 around -inf 25.6%

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

       1. +-commutative 25.6%

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

       2. mul-1-neg 25.6%

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

       3. sub-neg 25.6%

          \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

       4. *-commutative 25.6%

          \[\leadsto R \cdot \phi_2 - \color{blue}{\phi_1 \cdot R} \]

       5. cancel-sign-sub-inv 25.6%

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

       6. *-commutative 25.6%

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

       7. neg-mul-1 25.6%

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

       8. distribute-rgt-out 26.9%

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

       9. neg-mul-1 26.9%

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

       10. sub-neg 26.9%

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

   16. Simplified 26.9%

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

   if 1.25e172 < lambda2

   1. Initial program 45.1%

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

      1. hypot-def 95.0%

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

   3. Simplified 95.0%

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

   4. Taylor expanded in lambda2 around inf 55.2%

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

      1. *-commutative 55.2%

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

      2. *-commutative 55.2%

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

      3. *-commutative 55.2%

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

      4. associate-*l* 55.2%

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

      5. *-commutative 55.2%

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

      6. +-commutative 55.2%

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

   6. Simplified 55.2%

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

   7. Taylor expanded in phi1 around 0 59.1%

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

      1. *-commutative 59.1%

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

   9. Simplified 59.1%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.25 \cdot 10^{+172}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \cos \left(0.5 \cdot \phi_2\right)\right)\\ \end{array} \]

Alternative 7: 28.4% accurate, 54.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.18:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -0.18) (* R (- phi1)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -0.18) {
		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 <= (-0.18d0)) 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 <= -0.18) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -0.18:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -0.18)
		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 <= -0.18)
		tmp = R * -phi1;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -0.18], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -0.17999999999999999

   1. Initial program 48.2%

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

      1. hypot-def 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. Taylor expanded in phi1 around -inf 55.8%

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

      1. mul-1-neg 55.8%

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

      2. *-commutative 55.8%

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

      3. distribute-rgt-neg-in 55.8%

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

   6. Simplified 55.8%

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

   if -0.17999999999999999 < phi1

   1. Initial program 62.9%

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

      1. hypot-def 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. Taylor expanded in phi2 around inf 16.6%

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

      1. *-commutative 16.6%

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

   6. Simplified 16.6%

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

4. Final simplification 23.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.18:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]

Alternative 8: 29.5% accurate, 65.8× speedup

Math

\[\begin{array}{l} \\ R \cdot \left(\phi_2 - \phi_1\right) \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * (phi2 - phi1)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.1%

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

   1. hypot-def 96.6%

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

3. Simplified 96.6%

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

   1. add-cube-cbrt 95.4%

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

   2. pow3 95.3%

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

   3. *-commutative 95.3%

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

   4. div-inv 95.3%

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

   5. metadata-eval 95.3%

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

5. Applied egg-rr 95.3%

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

   1. *-commutative 95.3%

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

   2. +-commutative 95.3%

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

   3. distribute-rgt-in 95.3%

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

   4. cos-sum 98.5%

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

7. Applied egg-rr 98.5%

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

   1. fma-neg 98.5%

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

   2. *-commutative 98.5%

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

   3. *-commutative 98.5%

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

   4. *-commutative 98.5%

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

   5. *-commutative 98.5%

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

   6. distribute-rgt-neg-in 98.5%

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

9. Simplified 98.5%

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

    1. rem-cube-cbrt 99.8%

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

    2. expm1-log1p-u 54.0%

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

    3. expm1-udef 39.2%

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

11. Applied egg-rr 39.2%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(0.5 \cdot \phi_2\right) \cdot \left(-\sin \left(0.5 \cdot \phi_1\right)\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 54.0%

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

    2. expm1-log1p 99.8%

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

    3. *-commutative 99.8%

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

    4. distribute-rgt-neg-in 99.8%

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

    5. fma-neg 99.8%

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

    6. *-commutative 99.8%

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

    7. *-commutative 99.8%

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

13. Simplified 99.8%

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

14. Taylor expanded in phi1 around -inf 22.5%

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

    1. +-commutative 22.5%

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

    2. mul-1-neg 22.5%

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

    3. sub-neg 22.5%

       \[\leadsto \color{blue}{R \cdot \phi_2 - R \cdot \phi_1} \]

    4. *-commutative 22.5%

       \[\leadsto R \cdot \phi_2 - \color{blue}{\phi_1 \cdot R} \]

    5. cancel-sign-sub-inv 22.5%

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

    6. *-commutative 22.5%

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

    7. neg-mul-1 22.5%

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

    8. distribute-rgt-out 23.7%

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

    9. neg-mul-1 23.7%

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

    10. sub-neg 23.7%

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

16. Simplified 23.7%

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

17. Final simplification 23.7%

    \[\leadsto R \cdot \left(\phi_2 - \phi_1\right) \]

Alternative 9: 17.3% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r * phi2
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.1%

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

   1. hypot-def 96.6%

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

3. Simplified 96.6%

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

4. Taylor expanded in phi2 around inf 15.8%

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

   1. *-commutative 15.8%

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

6. Simplified 15.8%

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

7. Final simplification 15.8%

   \[\leadsto R \cdot \phi_2 \]

Reproduce

herbie shell --seed 2023311 
(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))))))