Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 99.9%

Time: 23.8s

Alternatives: 23

Speedup: 3.0×

• 31.2% 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, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. hypot-define 94.9%

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

3. Simplified 94.9%

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

   1. add-cube-cbrt 94.6%

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

   2. pow3 94.6%

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

   3. div-inv 94.6%

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

   4. metadata-eval 94.6%

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

6. Applied egg-rr 94.6%

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

   1. *-commutative 94.6%

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

   2. +-commutative 94.6%

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

   3. distribute-rgt-in 94.6%

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

   4. *-commutative 94.6%

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

   5. cos-sum 99.5%

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

   6. *-commutative 99.5%

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

   7. *-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

   1. rem-cube-cbrt 99.9%

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

   2. cancel-sign-sub-inv 99.9%

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

   3. *-commutative 99.9%

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

   4. fma-define 99.9%

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

   5. *-commutative 99.9%

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

10. Applied egg-rr 99.9%

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

11. Final simplification 99.9%

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

Alternative 2: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -5e-47)
   (*
    R
    (hypot (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0))) (- phi1 phi2)))
   (*
    R
    (hypot
     (*
      lambda2
      (-
       (* (sin (* phi2 0.5)) (sin (* 0.5 phi1)))
       (* (cos (* 0.5 phi1)) (cos (* phi2 0.5)))))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-47) {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda2 * ((sin((phi2 * 0.5)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * cos((phi2 * 0.5))))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -5e-47) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda2 * ((Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1))) - (Math.cos((0.5 * phi1)) * Math.cos((phi2 * 0.5))))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -5e-47:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda2 * ((math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1))) - (math.cos((0.5 * phi1)) * math.cos((phi2 * 0.5))))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -5e-47)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda2 * Float64(Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))) - Float64(cos(Float64(0.5 * phi1)) * cos(Float64(phi2 * 0.5))))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -5e-47)
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda2 * ((sin((phi2 * 0.5)) * sin((0.5 * phi1))) - (cos((0.5 * phi1)) * cos((phi2 * 0.5))))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -5e-47], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda2 * N[(N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -5.00000000000000011e-47

   1. Initial program 61.8%

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

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   if -5.00000000000000011e-47 < lambda1

   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-define 94.0%

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

   3. Simplified 94.0%

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

      1. add-cube-cbrt 93.8%

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

      2. pow3 93.8%

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

      3. div-inv 93.8%

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

      4. metadata-eval 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. *-commutative 93.8%

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

      2. +-commutative 93.8%

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

      3. distribute-rgt-in 93.8%

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

      4. *-commutative 93.8%

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

      5. cos-sum 99.6%

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

      6. *-commutative 99.6%

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

      7. *-commutative 99.6%

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

   8. Applied egg-rr 99.6%

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

   9. Taylor expanded in lambda1 around 0 86.6%

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

       1. mul-1-neg 86.6%

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

       2. *-commutative 86.6%

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

       3. distribute-rgt-neg-in 86.6%

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

   11. Simplified 86.6%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -5 \cdot 10^{-47}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_2 \cdot \left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right) - \cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. hypot-define 94.9%

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

3. Simplified 94.9%

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

   1. add-cube-cbrt 94.6%

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

   2. pow3 94.6%

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

   3. div-inv 94.6%

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

   4. metadata-eval 94.6%

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

6. Applied egg-rr 94.6%

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

   1. *-commutative 94.6%

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

   2. +-commutative 94.6%

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

   3. distribute-rgt-in 94.6%

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

   4. *-commutative 94.6%

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

   5. cos-sum 99.5%

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

   6. *-commutative 99.5%

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

   7. *-commutative 99.5%

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

8. Applied egg-rr 99.5%

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

9. Taylor expanded in lambda1 around 0 99.9%

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

    1. +-commutative 99.9%

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

    2. *-commutative 99.9%

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

    3. mul-1-neg 99.9%

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

    4. *-commutative 99.9%

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

    5. distribute-rgt-neg-in 99.9%

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

    6. distribute-lft-in 99.9%

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

    7. sub-neg 99.9%

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

11. Simplified 99.9%

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

12. Final simplification 99.9%

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

Alternative 4: 81.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1e-14)
   (* R (hypot (* lambda1 (cos (* 0.5 (+ phi2 phi1)))) (- phi1 phi2)))
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda2)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e-14) {
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * hypot((cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1e-14) {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1e-14:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * -lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1e-14)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(0.5 * Float64(phi2 + phi1)))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(-lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1e-14)
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	else
		tmp = R * hypot((cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1e-14], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * (-lambda2)), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.99999999999999999e-15

   1. Initial program 63.4%

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

      1. hypot-define 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in lambda1 around inf 82.9%

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

      1. *-commutative 82.9%

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

      2. *-commutative 82.9%

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

      3. remove-double-neg 82.9%

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

      4. mul-1-neg 82.9%

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

      5. sub-neg 82.9%

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

      6. *-commutative 82.9%

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

      7. sub-neg 82.9%

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

      8. mul-1-neg 82.9%

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

      9. remove-double-neg 82.9%

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

      10. +-commutative 82.9%

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

      11. *-lft-identity 82.9%

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

      12. metadata-eval 82.9%

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

      13. cancel-sign-sub-inv 82.9%

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

      14. sub-neg 82.9%

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

      15. mul-1-neg 82.9%

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

      16. remove-double-neg 82.9%

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

   7. Simplified 82.9%

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

   if 9.99999999999999999e-15 < lambda2

   1. Initial program 58.5%

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

      1. hypot-define 90.5%

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

   3. Simplified 90.5%

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

      1. add-cube-cbrt 90.2%

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

      2. pow3 90.2%

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

      3. div-inv 90.2%

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

      4. metadata-eval 90.2%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in phi2 around 0 82.9%

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

   8. Taylor expanded in lambda1 around 0 72.0%

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

      1. associate-*r* 72.0%

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

      2. mul-1-neg 72.0%

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

   10. Simplified 72.0%

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

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 10^{-14}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 6.5 \cdot 10^{-49}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 6.5e-49)
   (* R (hypot (* lambda1 (cos (* 0.5 (+ phi2 phi1)))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 6.5e-49) {
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 6.5e-49) {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 6.5e-49:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * (phi2 + phi1)))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 6.5e-49)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(0.5 * Float64(phi2 + phi1)))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 6.5e-49)
		tmp = R * hypot((lambda1 * cos((0.5 * (phi2 + phi1)))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 6.5e-49], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(0.5 * N[(phi2 + phi1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 6.49999999999999968e-49

   1. Initial program 63.2%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in lambda1 around inf 82.3%

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

      1. *-commutative 82.3%

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

      2. *-commutative 82.3%

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

      3. remove-double-neg 82.3%

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

      4. mul-1-neg 82.3%

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

      5. sub-neg 82.3%

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

      6. *-commutative 82.3%

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

      7. sub-neg 82.3%

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

      8. mul-1-neg 82.3%

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

      9. remove-double-neg 82.3%

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

      10. +-commutative 82.3%

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

      11. *-lft-identity 82.3%

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

      12. metadata-eval 82.3%

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

      13. cancel-sign-sub-inv 82.3%

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

      14. sub-neg 82.3%

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

      15. mul-1-neg 82.3%

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

      16. remove-double-neg 82.3%

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

   7. Simplified 82.3%

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

   if 6.49999999999999968e-49 < lambda2

   1. Initial program 59.4%

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

      1. hypot-define 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in phi2 around 0 84.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 6.5 \cdot 10^{-49}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \left(\phi_2 + \phi_1\right)\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.2 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 8.2e-21)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.2e-21) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 8.2e-21) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 8.2e-21:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 8.2e-21)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 8.2e-21)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 8.2e-21], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 8.19999999999999988e-21

   1. Initial program 60.1%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in phi2 around 0 92.5%

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

   if 8.19999999999999988e-21 < phi2

   1. Initial program 67.1%

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

      1. hypot-define 89.7%

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

   3. Simplified 89.7%

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

   5. Taylor expanded in phi1 around 0 89.8%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 8.2 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.2e-11)
   (* R (hypot (* lambda1 (cos (* phi2 0.5))) (- phi1 phi2)))
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda2)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.2e-11) {
		tmp = R * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot((cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.2e-11) {
		tmp = R * Math.hypot((lambda1 * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.2e-11:
		tmp = R * math.hypot((lambda1 * math.cos((phi2 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * -lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.2e-11)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(-lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.2e-11)
		tmp = R * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot((cos((0.5 * phi1)) * -lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.2e-11], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * (-lambda2)), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.2000000000000001e-11

   1. Initial program 63.4%

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

      1. hypot-define 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in phi1 around 0 92.6%

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

   6. Taylor expanded in lambda1 around inf 80.9%

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

      1. *-commutative 80.9%

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

   8. Simplified 80.9%

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

   if 1.2000000000000001e-11 < lambda2

   1. Initial program 58.5%

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

      1. hypot-define 90.5%

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

   3. Simplified 90.5%

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

      1. add-cube-cbrt 90.2%

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

      2. pow3 90.2%

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

      3. div-inv 90.2%

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

      4. metadata-eval 90.2%

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

   6. Applied egg-rr 90.2%

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

   7. Taylor expanded in phi2 around 0 82.9%

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

   8. Taylor expanded in lambda1 around 0 72.0%

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

      1. associate-*r* 72.0%

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

      2. mul-1-neg 72.0%

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

   10. Simplified 72.0%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.2 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(-\lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4 \cdot 10^{-187}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 4e-187)
   (* R (hypot (* lambda1 (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4e-187) {
		tmp = R * hypot((lambda1 * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 4e-187) {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 4e-187:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 4e-187)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 4e-187)
		tmp = R * hypot((lambda1 * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 4e-187], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 4.0000000000000001e-187

   1. Initial program 60.9%

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

      1. hypot-define 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in lambda1 around inf 79.5%

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

      1. *-commutative 79.5%

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

      2. *-commutative 79.5%

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

      3. remove-double-neg 79.5%

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

      4. mul-1-neg 79.5%

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

      5. sub-neg 79.5%

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

      6. *-commutative 79.5%

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

      7. sub-neg 79.5%

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

      8. mul-1-neg 79.5%

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

      9. remove-double-neg 79.5%

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

      10. +-commutative 79.5%

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

      11. *-lft-identity 79.5%

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

      12. metadata-eval 79.5%

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

      13. cancel-sign-sub-inv 79.5%

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

      14. sub-neg 79.5%

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

      15. mul-1-neg 79.5%

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

      16. remove-double-neg 79.5%

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

   7. Simplified 79.5%

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

   8. Taylor expanded in phi2 around 0 75.4%

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

      1. *-commutative 75.4%

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

   10. Simplified 75.4%

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

   if 4.0000000000000001e-187 < lambda2

   1. Initial program 64.3%

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

      1. hypot-define 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in phi1 around 0 86.2%

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

   6. Taylor expanded in phi2 around 0 81.6%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 4 \cdot 10^{-187}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 80.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 1.4e-53)
   (* R (hypot (* lambda1 (cos (* phi2 0.5))) (- phi1 phi2)))
   (* R (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.4e-53) {
		tmp = R * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.4e-53) {
		tmp = R * Math.hypot((lambda1 * Math.cos((phi2 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.4e-53:
		tmp = R * math.hypot((lambda1 * math.cos((phi2 * 0.5))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.4e-53)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(phi2 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.4e-53)
		tmp = R * hypot((lambda1 * cos((phi2 * 0.5))), (phi1 - phi2));
	else
		tmp = R * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 1.4e-53], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.39999999999999993e-53

   1. Initial program 63.2%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi1 around 0 92.8%

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

   6. Taylor expanded in lambda1 around inf 80.7%

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

      1. *-commutative 80.7%

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

   8. Simplified 80.7%

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

   if 1.39999999999999993e-53 < lambda2

   1. Initial program 59.4%

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

      1. hypot-define 91.3%

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

   3. Simplified 91.3%

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

   5. Taylor expanded in phi1 around 0 82.8%

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

   6. Taylor expanded in phi2 around 0 76.1%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.4 \cdot 10^{-53}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 96.1% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. hypot-define 94.9%

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

3. Simplified 94.9%

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

5. Final simplification 94.9%

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

Alternative 11: 70.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.35 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.35e+28)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.35e+28) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.35e+28) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.35e+28:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.35e+28)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 2.35e+28)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.35e+28], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.34999999999999983e28

   1. Initial program 61.6%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi1 around 0 90.0%

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

   6. Taylor expanded in phi2 around 0 48.1%

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

      1. unpow2 48.1%

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

      2. unpow2 48.1%

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

      3. hypot-define 67.0%

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

   8. Simplified 67.0%

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

   if 2.34999999999999983e28 < phi2

   1. Initial program 64.5%

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

      1. hypot-define 90.4%

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

   3. Simplified 90.4%

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

   5. Taylor expanded in phi2 around inf 67.4%

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

      1. mul-1-neg 67.4%

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

      2. unsub-neg 67.4%

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

   7. Simplified 67.4%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.35 \cdot 10^{+28}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-32}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 + \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.6e-32)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (+ lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.6e-32) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 + lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.6e-32) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 + lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.6e-32:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 + lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.6e-32)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 + lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.6e-32)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 + lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.6e-32], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 + lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.5999999999999997e-32

   1. Initial program 56.1%

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

      1. hypot-define 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in phi1 around 0 76.7%

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

   6. Taylor expanded in phi2 around 0 47.1%

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

      1. unpow2 47.1%

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

      2. unpow2 47.1%

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

      3. hypot-define 70.3%

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

   8. Simplified 70.3%

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

   if -2.5999999999999997e-32 < phi1

   1. Initial program 64.0%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

      1. add-cube-cbrt 96.9%

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

      2. pow3 96.9%

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

      3. div-inv 96.9%

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

      4. metadata-eval 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in phi2 around 0 90.0%

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

      1. add-cube-cbrt 88.9%

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

      2. pow3 88.9%

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

      3. sub-neg 88.9%

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

      4. add-sqr-sqrt 48.8%

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

      5. sqrt-unprod 79.7%

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

      6. sqr-neg 79.7%

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

      7. sqrt-unprod 40.2%

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

      8. add-sqr-sqrt 88.6%

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

      9. rem-cube-cbrt 88.6%

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

   9. Applied egg-rr 88.6%

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

   10. Taylor expanded in phi1 around 0 54.5%

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

       1. unpow2 54.5%

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

       2. unpow2 54.5%

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

       3. hypot-define 73.5%

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

       4. +-commutative 73.5%

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

   12. Simplified 73.5%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{-32}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 + \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 71.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-32}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.4e-32)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.4e-32) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.4e-32) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.4e-32:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.4e-32)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.4e-32)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.4e-32], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.4000000000000001e-32

   1. Initial program 56.1%

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

      1. hypot-define 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in phi1 around 0 76.7%

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

   6. Taylor expanded in phi2 around 0 47.1%

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

      1. unpow2 47.1%

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

      2. unpow2 47.1%

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

      3. hypot-define 70.3%

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

   8. Simplified 70.3%

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

   if -2.4000000000000001e-32 < phi1

   1. Initial program 64.0%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

      1. add-cube-cbrt 96.9%

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

      2. pow3 96.9%

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

      3. div-inv 96.9%

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

      4. metadata-eval 96.9%

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

   6. Applied egg-rr 96.9%

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

   7. Taylor expanded in phi2 around 0 90.0%

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

   8. Taylor expanded in phi1 around 0 54.5%

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

      1. unpow2 54.5%

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

      2. unpow2 54.5%

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

      3. hypot-define 73.8%

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

   10. Simplified 73.8%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-32}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 86.0% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. hypot-define 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in phi1 around 0 90.2%

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

6. Taylor expanded in phi2 around 0 84.7%

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

7. Final simplification 84.7%

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

Alternative 15: 30.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-299}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-156}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* lambda1 (- (* lambda2 (/ R lambda1)) R))))
   (if (<= phi1 -8e+55)
     (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
     (if (<= phi1 -3e-299)
       t_0
       (if (<= phi1 3.7e-271)
         (* R phi2)
         (if (<= phi1 9.5e-156) t_0 (* R (* phi2 (- 1.0 (/ phi1 phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R);
	double tmp;
	if (phi1 <= -8e+55) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (phi1 <= -3e-299) {
		tmp = t_0;
	} else if (phi1 <= 3.7e-271) {
		tmp = R * phi2;
	} else if (phi1 <= 9.5e-156) {
		tmp = t_0;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = lambda1 * ((lambda2 * (r / lambda1)) - r)
    if (phi1 <= (-8d+55)) then
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else if (phi1 <= (-3d-299)) then
        tmp = t_0
    else if (phi1 <= 3.7d-271) then
        tmp = r * phi2
    else if (phi1 <= 9.5d-156) then
        tmp = t_0
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R);
	double tmp;
	if (phi1 <= -8e+55) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else if (phi1 <= -3e-299) {
		tmp = t_0;
	} else if (phi1 <= 3.7e-271) {
		tmp = R * phi2;
	} else if (phi1 <= 9.5e-156) {
		tmp = t_0;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R)
	tmp = 0
	if phi1 <= -8e+55:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	elif phi1 <= -3e-299:
		tmp = t_0
	elif phi1 <= 3.7e-271:
		tmp = R * phi2
	elif phi1 <= 9.5e-156:
		tmp = t_0
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(lambda1 * Float64(Float64(lambda2 * Float64(R / lambda1)) - R))
	tmp = 0.0
	if (phi1 <= -8e+55)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	elseif (phi1 <= -3e-299)
		tmp = t_0;
	elseif (phi1 <= 3.7e-271)
		tmp = Float64(R * phi2);
	elseif (phi1 <= 9.5e-156)
		tmp = t_0;
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R);
	tmp = 0.0;
	if (phi1 <= -8e+55)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	elseif (phi1 <= -3e-299)
		tmp = t_0;
	elseif (phi1 <= 3.7e-271)
		tmp = R * phi2;
	elseif (phi1 <= 9.5e-156)
		tmp = t_0;
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(lambda1 * N[(N[(lambda2 * N[(R / lambda1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8e+55], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -3e-299], t$95$0, If[LessEqual[phi1, 3.7e-271], N[(R * phi2), $MachinePrecision], If[LessEqual[phi1, 9.5e-156], t$95$0, N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -8.00000000000000008e55

   1. Initial program 47.3%

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

      1. hypot-define 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in phi1 around -inf 69.7%

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

      1. mul-1-neg 69.7%

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

      2. distribute-rgt-neg-in 69.7%

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

      3. mul-1-neg 69.7%

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

      4. unsub-neg 69.7%

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

   7. Simplified 69.7%

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

   if -8.00000000000000008e55 < phi1 < -2.99999999999999984e-299 or 3.70000000000000022e-271 < phi1 < 9.4999999999999994e-156

   1. Initial program 70.6%

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

      1. hypot-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in phi1 around 0 95.9%

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

   6. Taylor expanded in lambda1 around -inf 39.0%

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

      1. mul-1-neg 39.0%

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

      2. *-commutative 39.0%

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

      3. distribute-rgt-neg-in 39.0%

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

      4. +-commutative 39.0%

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

      5. mul-1-neg 39.0%

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

      6. unsub-neg 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-*r* 39.0%

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

      9. *-commutative 39.0%

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

   8. Simplified 39.0%

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

   9. Taylor expanded in phi2 around 0 32.7%

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

       1. associate-*r* 32.7%

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

       2. neg-mul-1 32.7%

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

       3. *-commutative 32.7%

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

       4. associate-/l* 33.0%

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

   11. Simplified 33.0%

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

   if -2.99999999999999984e-299 < phi1 < 3.70000000000000022e-271

   1. Initial program 73.0%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around inf 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   if 9.4999999999999994e-156 < phi1

   1. Initial program 58.7%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in phi2 around inf 16.4%

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

      1. mul-1-neg 16.4%

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

      2. unsub-neg 16.4%

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

   7. Simplified 16.4%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-299}:\\ \;\;\;\;\lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 9.5 \cdot 10^{-156}:\\ \;\;\;\;\lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 30.4% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{+55}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-299}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* lambda1 (- (* lambda2 (/ R lambda1)) R))))
   (if (<= phi1 -8e+55)
     (* phi1 (- (* R (/ phi2 phi1)) R))
     (if (<= phi1 -3e-299)
       t_0
       (if (<= phi1 3.7e-271)
         (* R phi2)
         (if (<= phi1 2.7e-155) t_0 (* R (* phi2 (- 1.0 (/ phi1 phi2))))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R);
	double tmp;
	if (phi1 <= -8e+55) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -3e-299) {
		tmp = t_0;
	} else if (phi1 <= 3.7e-271) {
		tmp = R * phi2;
	} else if (phi1 <= 2.7e-155) {
		tmp = t_0;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = lambda1 * ((lambda2 * (r / lambda1)) - r)
    if (phi1 <= (-8d+55)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-3d-299)) then
        tmp = t_0
    else if (phi1 <= 3.7d-271) then
        tmp = r * phi2
    else if (phi1 <= 2.7d-155) then
        tmp = t_0
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R);
	double tmp;
	if (phi1 <= -8e+55) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -3e-299) {
		tmp = t_0;
	} else if (phi1 <= 3.7e-271) {
		tmp = R * phi2;
	} else if (phi1 <= 2.7e-155) {
		tmp = t_0;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R)
	tmp = 0
	if phi1 <= -8e+55:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -3e-299:
		tmp = t_0
	elif phi1 <= 3.7e-271:
		tmp = R * phi2
	elif phi1 <= 2.7e-155:
		tmp = t_0
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(lambda1 * Float64(Float64(lambda2 * Float64(R / lambda1)) - R))
	tmp = 0.0
	if (phi1 <= -8e+55)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -3e-299)
		tmp = t_0;
	elseif (phi1 <= 3.7e-271)
		tmp = Float64(R * phi2);
	elseif (phi1 <= 2.7e-155)
		tmp = t_0;
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = lambda1 * ((lambda2 * (R / lambda1)) - R);
	tmp = 0.0;
	if (phi1 <= -8e+55)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -3e-299)
		tmp = t_0;
	elseif (phi1 <= 3.7e-271)
		tmp = R * phi2;
	elseif (phi1 <= 2.7e-155)
		tmp = t_0;
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(lambda1 * N[(N[(lambda2 * N[(R / lambda1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi1, -8e+55], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -3e-299], t$95$0, If[LessEqual[phi1, 3.7e-271], N[(R * phi2), $MachinePrecision], If[LessEqual[phi1, 2.7e-155], t$95$0, N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -8.00000000000000008e55

   1. Initial program 47.3%

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

      1. hypot-define 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in phi1 around 0 77.9%

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

   6. Taylor expanded in phi1 around -inf 69.7%

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

      1. associate-*r* 69.7%

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

      2. mul-1-neg 69.7%

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

      3. mul-1-neg 69.7%

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

      4. unsub-neg 69.7%

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

      5. associate-/l* 69.7%

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

   8. Simplified 69.7%

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

   if -8.00000000000000008e55 < phi1 < -2.99999999999999984e-299 or 3.70000000000000022e-271 < phi1 < 2.69999999999999981e-155

   1. Initial program 70.6%

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

      1. hypot-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in phi1 around 0 95.9%

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

   6. Taylor expanded in lambda1 around -inf 39.0%

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

      1. mul-1-neg 39.0%

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

      2. *-commutative 39.0%

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

      3. distribute-rgt-neg-in 39.0%

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

      4. +-commutative 39.0%

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

      5. mul-1-neg 39.0%

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

      6. unsub-neg 39.0%

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

      7. *-commutative 39.0%

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

      8. associate-*r* 39.0%

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

      9. *-commutative 39.0%

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

   8. Simplified 39.0%

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

   9. Taylor expanded in phi2 around 0 32.7%

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

       1. associate-*r* 32.7%

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

       2. neg-mul-1 32.7%

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

       3. *-commutative 32.7%

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

       4. associate-/l* 33.0%

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

   11. Simplified 33.0%

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

   if -2.99999999999999984e-299 < phi1 < 3.70000000000000022e-271

   1. Initial program 73.0%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around inf 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   if 2.69999999999999981e-155 < phi1

   1. Initial program 58.7%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in phi2 around inf 16.4%

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

      1. mul-1-neg 16.4%

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

      2. unsub-neg 16.4%

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

   7. Simplified 16.4%

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

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8 \cdot 10^{+55}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-299}:\\ \;\;\;\;\lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;\lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 30.5% accurate, 11.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.4 \cdot 10^{+55}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-299}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;\lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -8.4e+55)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi1 -3e-299)
     (* lambda1 (- (/ (* R lambda2) lambda1) R))
     (if (<= phi1 3.7e-271)
       (* R phi2)
       (if (<= phi1 2.7e-155)
         (* lambda1 (- (* lambda2 (/ R lambda1)) R))
         (* R (* phi2 (- 1.0 (/ phi1 phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8.4e+55) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -3e-299) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else if (phi1 <= 3.7e-271) {
		tmp = R * phi2;
	} else if (phi1 <= 2.7e-155) {
		tmp = lambda1 * ((lambda2 * (R / lambda1)) - R);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-8.4d+55)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-3d-299)) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    else if (phi1 <= 3.7d-271) then
        tmp = r * phi2
    else if (phi1 <= 2.7d-155) then
        tmp = lambda1 * ((lambda2 * (r / lambda1)) - r)
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -8.4e+55) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -3e-299) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else if (phi1 <= 3.7e-271) {
		tmp = R * phi2;
	} else if (phi1 <= 2.7e-155) {
		tmp = lambda1 * ((lambda2 * (R / lambda1)) - R);
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -8.4e+55:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -3e-299:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	elif phi1 <= 3.7e-271:
		tmp = R * phi2
	elif phi1 <= 2.7e-155:
		tmp = lambda1 * ((lambda2 * (R / lambda1)) - R)
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -8.4e+55)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -3e-299)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	elseif (phi1 <= 3.7e-271)
		tmp = Float64(R * phi2);
	elseif (phi1 <= 2.7e-155)
		tmp = Float64(lambda1 * Float64(Float64(lambda2 * Float64(R / lambda1)) - R));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -8.4e+55)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -3e-299)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	elseif (phi1 <= 3.7e-271)
		tmp = R * phi2;
	elseif (phi1 <= 2.7e-155)
		tmp = lambda1 * ((lambda2 * (R / lambda1)) - R);
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -8.4e+55], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -3e-299], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 3.7e-271], N[(R * phi2), $MachinePrecision], If[LessEqual[phi1, 2.7e-155], N[(lambda1 * N[(N[(lambda2 * N[(R / lambda1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if phi1 < -8.4000000000000002e55

   1. Initial program 47.3%

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

      1. hypot-define 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in phi1 around 0 77.9%

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

   6. Taylor expanded in phi1 around -inf 69.7%

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

      1. associate-*r* 69.7%

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

      2. mul-1-neg 69.7%

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

      3. mul-1-neg 69.7%

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

      4. unsub-neg 69.7%

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

      5. associate-/l* 69.7%

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

   8. Simplified 69.7%

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

   if -8.4000000000000002e55 < phi1 < -2.99999999999999984e-299

   1. Initial program 69.6%

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

      1. hypot-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in phi1 around 0 94.6%

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

   6. Taylor expanded in lambda1 around -inf 41.0%

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

      1. mul-1-neg 41.0%

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

      2. *-commutative 41.0%

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

      3. distribute-rgt-neg-in 41.0%

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

      4. +-commutative 41.0%

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

      5. mul-1-neg 41.0%

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

      6. unsub-neg 41.0%

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

      7. *-commutative 41.0%

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

      8. associate-*r* 41.0%

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

      9. *-commutative 41.0%

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

   8. Simplified 41.0%

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

   9. Taylor expanded in phi2 around 0 33.5%

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

   if -2.99999999999999984e-299 < phi1 < 3.70000000000000022e-271

   1. Initial program 73.0%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi2 around inf 51.0%

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

      1. *-commutative 51.0%

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

   7. Simplified 51.0%

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

   if 3.70000000000000022e-271 < phi1 < 2.69999999999999981e-155

   1. Initial program 73.6%

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

      1. hypot-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in phi1 around 0 99.9%

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

   6. Taylor expanded in lambda1 around -inf 33.2%

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

      1. mul-1-neg 33.2%

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

      2. *-commutative 33.2%

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

      3. distribute-rgt-neg-in 33.2%

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

      4. +-commutative 33.2%

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

      5. mul-1-neg 33.2%

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

      6. unsub-neg 33.2%

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

      7. *-commutative 33.2%

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

      8. associate-*r* 33.2%

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

      9. *-commutative 33.2%

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

   8. Simplified 33.2%

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

   9. Taylor expanded in phi2 around 0 30.3%

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

       1. associate-*r* 30.3%

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

       2. neg-mul-1 30.3%

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

       3. *-commutative 30.3%

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

       4. associate-/l* 32.5%

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

   11. Simplified 32.5%

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

   if 2.69999999999999981e-155 < phi1

   1. Initial program 58.7%

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

      1. hypot-define 94.3%

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

   3. Simplified 94.3%

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

   5. Taylor expanded in phi2 around inf 16.4%

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

      1. mul-1-neg 16.4%

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

      2. unsub-neg 16.4%

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

   7. Simplified 16.4%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.4 \cdot 10^{+55}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -3 \cdot 10^{-299}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq 3.7 \cdot 10^{-271}:\\ \;\;\;\;R \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 2.7 \cdot 10^{-155}:\\ \;\;\;\;\lambda_1 \cdot \left(\lambda_2 \cdot \frac{R}{\lambda_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.3% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-206}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.05e-206) (* phi1 (- R)) (* R (* phi2 (- 1.0 (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-206) {
		tmp = phi1 * -R;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.05d-206) then
        tmp = phi1 * -r
    else
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.05e-206) {
		tmp = phi1 * -R;
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.05e-206:
		tmp = phi1 * -R
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.05e-206)
		tmp = Float64(phi1 * Float64(-R));
	else
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.05e-206)
		tmp = phi1 * -R;
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.05e-206], N[(phi1 * (-R)), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.05000000000000005e-206

   1. Initial program 59.2%

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

      1. hypot-define 96.5%

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

   3. Simplified 96.5%

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

   5. Taylor expanded in phi1 around -inf 15.5%

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

      1. mul-1-neg 15.5%

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

      2. *-commutative 15.5%

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

      3. distribute-rgt-neg-in 15.5%

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

   7. Simplified 15.5%

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

   if 1.05000000000000005e-206 < phi2

   1. Initial program 66.3%

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

      1. hypot-define 92.7%

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

   3. Simplified 92.7%

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

   5. Taylor expanded in phi2 around inf 44.5%

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

      1. mul-1-neg 44.5%

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

      2. unsub-neg 44.5%

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

   7. Simplified 44.5%

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

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.05 \cdot 10^{-206}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 28.9% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.05 \cdot 10^{+86}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.05e+86) (* phi1 (- R)) (* phi2 (- R (* R (/ phi1 phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.05e+86) {
		tmp = phi1 * -R;
	} else {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.05d+86)) then
        tmp = phi1 * -r
    else
        tmp = phi2 * (r - (r * (phi1 / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.05e+86) {
		tmp = phi1 * -R;
	} else {
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.05e+86:
		tmp = phi1 * -R
	else:
		tmp = phi2 * (R - (R * (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.05e+86)
		tmp = Float64(phi1 * Float64(-R));
	else
		tmp = Float64(phi2 * Float64(R - Float64(R * Float64(phi1 / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.05e+86)
		tmp = phi1 * -R;
	else
		tmp = phi2 * (R - (R * (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.05e+86], N[(phi1 * (-R)), $MachinePrecision], N[(phi2 * N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.05e86

   1. Initial program 47.3%

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

      1. hypot-define 86.4%

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

   3. Simplified 86.4%

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

   5. Taylor expanded in phi1 around -inf 67.8%

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

      1. mul-1-neg 67.8%

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

      2. *-commutative 67.8%

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

      3. distribute-rgt-neg-in 67.8%

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

   7. Simplified 67.8%

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

   if -2.05e86 < phi1

   1. Initial program 65.2%

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

      1. hypot-define 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in phi1 around 0 92.6%

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

   6. Taylor expanded in phi2 around inf 19.7%

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

      1. mul-1-neg 19.7%

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

      2. unsub-neg 19.7%

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

      3. associate-/l* 21.1%

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

   8. Simplified 21.1%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.05 \cdot 10^{+86}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - R \cdot \frac{\phi_1}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 29.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.5e+105) (* phi1 (- R)) (* phi2 (- R (* phi1 (/ R phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e+105) {
		tmp = phi1 * -R;
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.5d+105)) then
        tmp = phi1 * -r
    else
        tmp = phi2 * (r - (phi1 * (r / phi2)))
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.5e+105) {
		tmp = phi1 * -R;
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.5e+105:
		tmp = phi1 * -R
	else:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.5e+105)
		tmp = Float64(phi1 * Float64(-R));
	else
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.5e+105)
		tmp = phi1 * -R;
	else
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.5e+105], N[(phi1 * (-R)), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.49999999999999991e105

   1. Initial program 45.0%

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

      1. hypot-define 86.1%

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

   3. Simplified 86.1%

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

   5. Taylor expanded in phi1 around -inf 72.1%

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

      1. mul-1-neg 72.1%

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

      2. *-commutative 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

   7. Simplified 72.1%

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

   if -3.49999999999999991e105 < phi1

   1. Initial program 65.0%

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

      1. hypot-define 96.3%

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

   3. Simplified 96.3%

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

   5. Taylor expanded in phi2 around inf 20.1%

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

      1. mul-1-neg 20.1%

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

      2. unsub-neg 20.1%

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

      3. *-commutative 20.1%

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

      4. associate-/l* 21.8%

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

   7. Simplified 21.8%

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

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.5 \cdot 10^{+105}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 30.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.06 \cdot 10^{+101}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.06e+101)
   (* R (* phi1 (+ (/ phi2 phi1) -1.0)))
   (* phi2 (- R (* phi1 (/ R phi2))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.06e+101) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = phi2 * (R - (phi1 * (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 <= (-1.06d+101)) then
        tmp = r * (phi1 * ((phi2 / phi1) + (-1.0d0)))
    else
        tmp = phi2 * (r - (phi1 * (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 <= -1.06e+101) {
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	} else {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.06e+101:
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0))
	else:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.06e+101)
		tmp = Float64(R * Float64(phi1 * Float64(Float64(phi2 / phi1) + -1.0)));
	else
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.06e+101)
		tmp = R * (phi1 * ((phi2 / phi1) + -1.0));
	else
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.06e+101], N[(R * N[(phi1 * N[(N[(phi2 / phi1), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.06e101

   1. Initial program 44.3%

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

      1. hypot-define 85.0%

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

   3. Simplified 85.0%

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

   5. Taylor expanded in phi1 around -inf 71.7%

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

      1. mul-1-neg 71.7%

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

      2. distribute-rgt-neg-in 71.7%

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

      3. mul-1-neg 71.7%

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

      4. unsub-neg 71.7%

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

   7. Simplified 71.7%

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

   if -1.06e101 < phi1

   1. Initial program 65.4%

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

      1. hypot-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in phi2 around inf 19.8%

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

      1. mul-1-neg 19.8%

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

      2. unsub-neg 19.8%

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

      3. *-commutative 19.8%

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

      4. associate-/l* 21.6%

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

   7. Simplified 21.6%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.06 \cdot 10^{+101}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(\frac{\phi_2}{\phi_1} + -1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 27.8% accurate, 36.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.5e-32) (* phi1 (- R)) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.5e-32) {
		tmp = phi1 * -R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-2.5d-32)) then
        tmp = phi1 * -r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.5e-32) {
		tmp = phi1 * -R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.5e-32:
		tmp = phi1 * -R
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.5e-32)
		tmp = Float64(phi1 * Float64(-R));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.5e-32)
		tmp = phi1 * -R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.5e-32], N[(phi1 * (-R)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.5e-32

   1. Initial program 56.1%

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

      1. hypot-define 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in phi1 around -inf 53.4%

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

      1. mul-1-neg 53.4%

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

      2. *-commutative 53.4%

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

      3. distribute-rgt-neg-in 53.4%

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

   7. Simplified 53.4%

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

   if -2.5e-32 < phi1

   1. Initial program 64.0%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in phi2 around inf 20.0%

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

      1. *-commutative 20.0%

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

   7. Simplified 20.0%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.5 \cdot 10^{-32}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 23: 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 62.2%

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

   1. hypot-define 94.9%

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

3. Simplified 94.9%

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

5. Taylor expanded in phi2 around inf 17.6%

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

   1. *-commutative 17.6%

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

7. Simplified 17.6%

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

8. Final simplification 17.6%

   \[\leadsto R \cdot \phi_2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024085 
(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))))))