Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.9% → 98.6%

Time: 27.8s

Alternatives: 17

Speedup: 3.0×

• 30.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Julia

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

Wolfram

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

Derivation

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 95.0%

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

3. Simplified 95.0%

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

   1. add-cube-cbrt 93.9%

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

   2. pow3 93.9%

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

   3. *-commutative 93.9%

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

   4. div-inv 93.9%

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

   5. metadata-eval 93.9%

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

6. Applied egg-rr 93.9%

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

   1. *-commutative 93.9%

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

   2. +-commutative 93.9%

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

   3. distribute-rgt-in 93.9%

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

   4. *-commutative 93.9%

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

   5. cos-sum 98.7%

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

   6. *-commutative 98.7%

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

   7. *-commutative 98.7%

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

8. Applied egg-rr 98.7%

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

9. Final simplification 98.7%

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

Alternative 2: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+211}:\\ \;\;\;\;e^{\log \left(R \cdot \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= (- lambda1 lambda2) -1e+211)
   (exp
    (log
     (*
      R
      (hypot
       (*
        (-
         (* (cos (* phi2 0.5)) (cos (* 0.5 phi1)))
         (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))
        (- lambda1 lambda2))
       (- phi1 phi2)))))
   (*
    R
    (hypot
     (* (- lambda1 lambda2) (cos (/ (+ phi2 phi1) 2.0)))
     (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+211) {
		tmp = exp(log((R * hypot((((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))));
	} else {
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if ((lambda1 - lambda2) <= -1e+211) {
		tmp = Math.exp(Math.log((R * Math.hypot((((Math.cos((phi2 * 0.5)) * Math.cos((0.5 * phi1))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))));
	} else {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if (lambda1 - lambda2) <= -1e+211:
		tmp = math.exp(math.log((R * math.hypot((((math.cos((phi2 * 0.5)) * math.cos((0.5 * phi1))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))))
	else:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos(((phi2 + phi1) / 2.0))), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (Float64(lambda1 - lambda2) <= -1e+211)
		tmp = exp(log(Float64(R * hypot(Float64(Float64(Float64(cos(Float64(phi2 * 0.5)) * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi2 + phi1) / 2.0))), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if ((lambda1 - lambda2) <= -1e+211)
		tmp = exp(log((R * hypot((((cos((phi2 * 0.5)) * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * (lambda1 - lambda2)), (phi1 - phi2)))));
	else
		tmp = R * hypot(((lambda1 - lambda2) * cos(((phi2 + phi1) / 2.0))), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[N[(lambda1 - lambda2), $MachinePrecision], -1e+211], N[Exp[N[Log[N[(R * N[Sqrt[N[(N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 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. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -9.9999999999999996e210

   1. Initial program 43.0%

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

      1. hypot-define 87.5%

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

   3. Simplified 87.5%

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

      1. add-cube-cbrt 86.8%

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

      2. pow3 86.9%

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

      3. *-commutative 86.9%

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

      4. div-inv 86.9%

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

      5. metadata-eval 86.9%

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

   6. Applied egg-rr 86.9%

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

      1. add-exp-log 44.8%

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

      2. rem-cube-cbrt 44.8%

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

      3. *-commutative 44.8%

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

      4. *-commutative 44.8%

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

   8. Applied egg-rr 44.8%

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

      1. *-commutative 86.9%

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

      2. +-commutative 86.9%

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

      3. distribute-rgt-in 86.9%

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

      4. *-commutative 86.9%

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

      5. cos-sum 99.0%

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

      6. *-commutative 99.0%

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

      7. *-commutative 99.0%

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

   10. Applied egg-rr 48.0%

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

   if -9.9999999999999996e210 < (-.f64 lambda1 lambda2)

   1. Initial program 62.1%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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
3. Recombined 2 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+211}:\\ \;\;\;\;e^{\log \left(R \cdot \mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_2 + \phi_1}{2}\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 7.2e-9)
   (* R (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) (- phi1 phi2)))
   (* R (hypot (* (cos (* phi2 0.5)) (- lambda1 lambda2)) (- phi1 phi2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.2e-9) {
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 7.2e-9) {
		tmp = R * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), (phi1 - phi2));
	} else {
		tmp = R * Math.hypot((Math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 7.2e-9:
		tmp = R * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), (phi1 - phi2))
	else:
		tmp = R * math.hypot((math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 7.2e-9)
		tmp = Float64(R * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(phi2 * 0.5)) * Float64(lambda1 - lambda2)), Float64(phi1 - phi2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 7.2e-9)
		tmp = R * hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), (phi1 - phi2));
	else
		tmp = R * hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), (phi1 - phi2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 7.2e-9], 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[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.2e-9

   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 96.9%

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

   3. Simplified 96.9%

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

   5. Taylor expanded in phi2 around 0 91.3%

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

   if 7.2e-9 < phi2

   1. Initial program 51.5%

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

      1. hypot-define 88.6%

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

   3. Simplified 88.6%

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

   5. Taylor expanded in phi1 around 0 88.6%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{-9}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -4.6 \cdot 10^{+101}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -4.6e+101)
   (* 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 (lambda1 <= -4.6e+101) {
		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 (lambda1 <= -4.6e+101) {
		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 lambda1 <= -4.6e+101:
		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 (lambda1 <= -4.6e+101)
		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 (lambda1 <= -4.6e+101)
		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[lambda1, -4.6e+101], 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 lambda1 < -4.6000000000000003e101

   1. Initial program 59.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 0 86.9%

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

   6. Taylor expanded in lambda1 around inf 80.4%

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

   if -4.6000000000000003e101 < lambda1

   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 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in phi2 around 0 88.6%

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

   6. Taylor expanded in phi1 around 0 83.7%

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

Alternative 5: 96.2% 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 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 95.0%

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

3. Simplified 95.0%

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

5. Final simplification 95.0%

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

Alternative 6: 90.8% accurate, 1.6× speedup

Math

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

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R (hypot (* (- lambda1 lambda2) (cos (* 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))), (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))), (phi1 - phi2));
}

Python

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

Julia

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

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := 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. 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 95.0%

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

3. Simplified 95.0%

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

5. Taylor expanded in phi2 around 0 88.3%

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

Alternative 7: 70.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{+32}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -2.15e+32)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (* R (hypot phi2 (- lambda1 lambda2)))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.15e+32) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.15e+32) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.15e+32:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.15e+32)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.15e+32)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -2.15e+32], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.1499999999999999e32

   1. Initial program 48.9%

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

      1. hypot-define 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in phi2 around 0 91.5%

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

   6. Taylor expanded in lambda2 around inf 83.1%

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

   7. Taylor expanded in phi1 around -inf 62.2%

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

      1. associate-*r* 62.2%

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

      2. neg-mul-1 62.2%

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

      3. mul-1-neg 62.2%

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

      4. unsub-neg 62.2%

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

      5. associate-/l* 62.2%

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

   9. Simplified 62.2%

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

   if -2.1499999999999999e32 < phi1

   1. Initial program 61.0%

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

      1. hypot-define 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in phi2 around 0 87.6%

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

   6. Taylor expanded in phi1 around 0 84.3%

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

   7. Taylor expanded in phi1 around 0 51.3%

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

      1. unpow2 51.3%

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

      2. unpow2 51.3%

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

      3. hypot-define 70.8%

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

   9. Simplified 70.8%

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

4. Final simplification 69.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.15 \cdot 10^{+32}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 85.2% 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 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 95.0%

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

3. Simplified 95.0%

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

5. Taylor expanded in phi2 around 0 88.3%

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

6. Taylor expanded in phi1 around 0 83.3%

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

Alternative 9: 27.8% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := R \cdot \left(-\lambda_1\right)\\ t_1 := \phi_1 \cdot \left(-R\right)\\ \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-132}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 3.1 \cdot 10^{-305}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{-275}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\phi_2 \leq 2.35 \cdot 10^{-224}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-127}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.04 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* R (- lambda1))) (t_1 (* phi1 (- R))))
   (if (<= phi2 -2e-132)
     t_1
     (if (<= phi2 3.1e-305)
       (* R lambda2)
       (if (<= phi2 8.5e-275)
         t_1
         (if (<= phi2 2.35e-224)
           t_0
           (if (<= phi2 2.3e-127)
             (* R lambda2)
             (if (<= phi2 1.04e-11) t_0 (* R phi2)))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double t_1 = phi1 * -R;
	double tmp;
	if (phi2 <= -2e-132) {
		tmp = t_1;
	} else if (phi2 <= 3.1e-305) {
		tmp = R * lambda2;
	} else if (phi2 <= 8.5e-275) {
		tmp = t_1;
	} else if (phi2 <= 2.35e-224) {
		tmp = t_0;
	} else if (phi2 <= 2.3e-127) {
		tmp = R * lambda2;
	} else if (phi2 <= 1.04e-11) {
		tmp = t_0;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = r * -lambda1
    t_1 = phi1 * -r
    if (phi2 <= (-2d-132)) then
        tmp = t_1
    else if (phi2 <= 3.1d-305) then
        tmp = r * lambda2
    else if (phi2 <= 8.5d-275) then
        tmp = t_1
    else if (phi2 <= 2.35d-224) then
        tmp = t_0
    else if (phi2 <= 2.3d-127) then
        tmp = r * lambda2
    else if (phi2 <= 1.04d-11) then
        tmp = t_0
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = R * -lambda1;
	double t_1 = phi1 * -R;
	double tmp;
	if (phi2 <= -2e-132) {
		tmp = t_1;
	} else if (phi2 <= 3.1e-305) {
		tmp = R * lambda2;
	} else if (phi2 <= 8.5e-275) {
		tmp = t_1;
	} else if (phi2 <= 2.35e-224) {
		tmp = t_0;
	} else if (phi2 <= 2.3e-127) {
		tmp = R * lambda2;
	} else if (phi2 <= 1.04e-11) {
		tmp = t_0;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = R * -lambda1
	t_1 = phi1 * -R
	tmp = 0
	if phi2 <= -2e-132:
		tmp = t_1
	elif phi2 <= 3.1e-305:
		tmp = R * lambda2
	elif phi2 <= 8.5e-275:
		tmp = t_1
	elif phi2 <= 2.35e-224:
		tmp = t_0
	elif phi2 <= 2.3e-127:
		tmp = R * lambda2
	elif phi2 <= 1.04e-11:
		tmp = t_0
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-lambda1))
	t_1 = Float64(phi1 * Float64(-R))
	tmp = 0.0
	if (phi2 <= -2e-132)
		tmp = t_1;
	elseif (phi2 <= 3.1e-305)
		tmp = Float64(R * lambda2);
	elseif (phi2 <= 8.5e-275)
		tmp = t_1;
	elseif (phi2 <= 2.35e-224)
		tmp = t_0;
	elseif (phi2 <= 2.3e-127)
		tmp = Float64(R * lambda2);
	elseif (phi2 <= 1.04e-11)
		tmp = t_0;
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = R * -lambda1;
	t_1 = phi1 * -R;
	tmp = 0.0;
	if (phi2 <= -2e-132)
		tmp = t_1;
	elseif (phi2 <= 3.1e-305)
		tmp = R * lambda2;
	elseif (phi2 <= 8.5e-275)
		tmp = t_1;
	elseif (phi2 <= 2.35e-224)
		tmp = t_0;
	elseif (phi2 <= 2.3e-127)
		tmp = R * lambda2;
	elseif (phi2 <= 1.04e-11)
		tmp = t_0;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(R * (-lambda1)), $MachinePrecision]}, Block[{t$95$1 = N[(phi1 * (-R)), $MachinePrecision]}, If[LessEqual[phi2, -2e-132], t$95$1, If[LessEqual[phi2, 3.1e-305], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi2, 8.5e-275], t$95$1, If[LessEqual[phi2, 2.35e-224], t$95$0, If[LessEqual[phi2, 2.3e-127], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi2, 1.04e-11], t$95$0, N[(R * phi2), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -2e-132 or 3.0999999999999998e-305 < phi2 < 8.49999999999999952e-275

   1. Initial program 58.8%

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

      1. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in phi1 around -inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   if -2e-132 < phi2 < 3.0999999999999998e-305 or 2.35000000000000001e-224 < phi2 < 2.30000000000000019e-127

   1. Initial program 61.7%

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

      1. hypot-define 99.9%

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

   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 0 99.9%

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

   6. Taylor expanded in phi1 around 0 91.8%

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

   7. Taylor expanded in lambda2 around inf 30.6%

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

   if 8.49999999999999952e-275 < phi2 < 2.35000000000000001e-224 or 2.30000000000000019e-127 < phi2 < 1.03999999999999993e-11

   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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in phi2 around 0 100.0%

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

   6. Taylor expanded in phi1 around 0 93.2%

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

   7. Taylor expanded in lambda1 around -inf 27.9%

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

      1. associate-*r* 27.9%

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

      2. neg-mul-1 27.9%

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

   9. Simplified 27.9%

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

   if 1.03999999999999993e-11 < phi2

   1. Initial program 52.3%

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

      1. hypot-define 88.8%

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

   3. Simplified 88.8%

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

   5. Taylor expanded in phi2 around inf 53.7%

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

      1. *-commutative 53.7%

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

   7. Simplified 53.7%

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

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2 \cdot 10^{-132}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.1 \cdot 10^{-305}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 8.5 \cdot 10^{-275}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.35 \cdot 10^{-224}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\phi_2 \leq 2.3 \cdot 10^{-127}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.04 \cdot 10^{-11}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 33.5% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-118}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-64}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 5000000:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -3.6e-118)
   (* phi1 (- R))
   (if (<= phi2 2.2e-64)
     (* lambda1 (- (/ (* R lambda2) lambda1) R))
     (if (<= phi2 5000000.0)
       (* R (* lambda2 (- 1.0 (/ 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 <= -3.6e-118) {
		tmp = phi1 * -R;
	} else if (phi2 <= 2.2e-64) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else if (phi2 <= 5000000.0) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} 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 <= (-3.6d-118)) then
        tmp = phi1 * -r
    else if (phi2 <= 2.2d-64) then
        tmp = lambda1 * (((r * lambda2) / lambda1) - r)
    else if (phi2 <= 5000000.0d0) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    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 <= -3.6e-118) {
		tmp = phi1 * -R;
	} else if (phi2 <= 2.2e-64) {
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	} else if (phi2 <= 5000000.0) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -3.6e-118:
		tmp = phi1 * -R
	elif phi2 <= 2.2e-64:
		tmp = lambda1 * (((R * lambda2) / lambda1) - R)
	elif phi2 <= 5000000.0:
		tmp = R * (lambda2 * (1.0 - (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 <= -3.6e-118)
		tmp = Float64(phi1 * Float64(-R));
	elseif (phi2 <= 2.2e-64)
		tmp = Float64(lambda1 * Float64(Float64(Float64(R * lambda2) / lambda1) - R));
	elseif (phi2 <= 5000000.0)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - 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 <= -3.6e-118)
		tmp = phi1 * -R;
	elseif (phi2 <= 2.2e-64)
		tmp = lambda1 * (((R * lambda2) / lambda1) - R);
	elseif (phi2 <= 5000000.0)
		tmp = R * (lambda2 * (1.0 - (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, -3.6e-118], N[(phi1 * (-R)), $MachinePrecision], If[LessEqual[phi2, 2.2e-64], N[(lambda1 * N[(N[(N[(R * lambda2), $MachinePrecision] / lambda1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 5000000.0], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -3.6000000000000002e-118

   1. Initial program 56.0%

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

      1. hypot-define 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in phi1 around -inf 12.5%

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

      1. mul-1-neg 12.5%

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

   7. Simplified 12.5%

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

   if -3.6000000000000002e-118 < phi2 < 2.2e-64

   1. Initial program 61.3%

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

      1. hypot-define 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 0 99.9%

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

   6. Taylor expanded in phi1 around 0 89.1%

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

   7. Taylor expanded in lambda1 around -inf 33.9%

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

      1. associate-*r* 33.9%

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

      2. mul-1-neg 33.9%

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

      3. associate-*r/ 33.9%

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

      4. mul-1-neg 33.9%

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

   9. Simplified 33.9%

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

   if 2.2e-64 < phi2 < 5e6

   1. Initial program 79.6%

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

      1. hypot-define 95.9%

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

   3. Simplified 95.9%

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

   5. Taylor expanded in phi2 around 0 86.5%

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

   6. Taylor expanded in phi1 around 0 82.1%

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

   7. Taylor expanded in lambda2 around inf 18.3%

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

      1. associate-*r/ 18.3%

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

      2. mul-1-neg 18.3%

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

   9. Simplified 18.3%

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

   if 5e6 < phi2

   1. Initial program 51.5%

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

      1. hypot-define 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in phi2 around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

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

   7. Simplified 60.9%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -3.6 \cdot 10^{-118}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-64}:\\ \;\;\;\;\lambda_1 \cdot \left(\frac{R \cdot \lambda_2}{\lambda_1} - R\right)\\ \mathbf{elif}\;\phi_2 \leq 5000000:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.7% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{+41}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -7.6 \cdot 10^{-84}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 8.5 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.6e+41)
   (* phi1 (- (* R (/ phi2 phi1)) R))
   (if (<= phi1 -7.6e-84)
     (* phi2 (- R (* phi1 (/ R phi2))))
     (if (<= phi1 8.5e-56)
       (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))
       (* R (* phi2 (- 1.0 (/ phi1 phi2))))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.6e+41) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -7.6e-84) {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	} else if (phi1 <= 8.5e-56) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} 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 <= (-1.6d+41)) then
        tmp = phi1 * ((r * (phi2 / phi1)) - r)
    else if (phi1 <= (-7.6d-84)) then
        tmp = phi2 * (r - (phi1 * (r / phi2)))
    else if (phi1 <= 8.5d-56) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    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 <= -1.6e+41) {
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	} else if (phi1 <= -7.6e-84) {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	} else if (phi1 <= 8.5e-56) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.6e+41:
		tmp = phi1 * ((R * (phi2 / phi1)) - R)
	elif phi1 <= -7.6e-84:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	elif phi1 <= 8.5e-56:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.6e+41)
		tmp = Float64(phi1 * Float64(Float64(R * Float64(phi2 / phi1)) - R));
	elseif (phi1 <= -7.6e-84)
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	elseif (phi1 <= 8.5e-56)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - 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 (phi1 <= -1.6e+41)
		tmp = phi1 * ((R * (phi2 / phi1)) - R);
	elseif (phi1 <= -7.6e-84)
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	elseif (phi1 <= 8.5e-56)
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.6e+41], N[(phi1 * N[(N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -7.6e-84], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, 8.5e-56], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.60000000000000005e41

   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 91.7%

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

   3. Simplified 91.7%

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

   5. Taylor expanded in phi2 around 0 91.7%

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

   6. Taylor expanded in lambda2 around inf 82.5%

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

   7. Taylor expanded in phi1 around -inf 64.2%

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

      1. associate-*r* 64.2%

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

      2. neg-mul-1 64.2%

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

      3. mul-1-neg 64.2%

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

      4. unsub-neg 64.2%

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

      5. associate-/l* 64.1%

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

   9. Simplified 64.1%

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

   if -1.60000000000000005e41 < phi1 < -7.59999999999999971e-84

   1. Initial program 68.4%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi2 around inf 33.4%

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

      1. mul-1-neg 33.4%

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

      2. unsub-neg 33.4%

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

      3. *-commutative 33.4%

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

      4. associate-/l* 36.5%

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

   7. Simplified 36.5%

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

   if -7.59999999999999971e-84 < phi1 < 8.49999999999999932e-56

   1. Initial program 66.8%

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

      1. hypot-define 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 0 87.6%

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

   6. Taylor expanded in phi1 around 0 87.6%

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

   7. Taylor expanded in lambda2 around inf 34.2%

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

      1. associate-*r/ 34.2%

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

      2. mul-1-neg 34.2%

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

   9. Simplified 34.2%

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

   if 8.49999999999999932e-56 < phi1

   1. Initial program 53.5%

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

      1. hypot-define 91.0%

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

   3. Simplified 91.0%

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

   5. Taylor expanded in phi2 around inf 11.0%

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

      1. mul-1-neg 11.0%

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

      2. unsub-neg 11.0%

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

   7. Simplified 11.0%

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

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.6 \cdot 10^{+41}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{elif}\;\phi_1 \leq -7.6 \cdot 10^{-84}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{elif}\;\phi_1 \leq 8.5 \cdot 10^{-56}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.9% accurate, 14.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \phi_1 \cdot \left(-R\right)\\ \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-129}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{-279}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 9.8 \cdot 10^{+70}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* phi1 (- R))))
   (if (<= phi2 -2.8e-129)
     t_0
     (if (<= phi2 4.5e-305)
       (* R lambda2)
       (if (<= phi2 1.8e-279)
         t_0
         (if (<= phi2 9.8e+70) (* R lambda2) (* R phi2)))))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * -R;
	double tmp;
	if (phi2 <= -2.8e-129) {
		tmp = t_0;
	} else if (phi2 <= 4.5e-305) {
		tmp = R * lambda2;
	} else if (phi2 <= 1.8e-279) {
		tmp = t_0;
	} else if (phi2 <= 9.8e+70) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: t_0
    real(8) :: tmp
    t_0 = phi1 * -r
    if (phi2 <= (-2.8d-129)) then
        tmp = t_0
    else if (phi2 <= 4.5d-305) then
        tmp = r * lambda2
    else if (phi2 <= 1.8d-279) then
        tmp = t_0
    else if (phi2 <= 9.8d+70) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = phi1 * -R;
	double tmp;
	if (phi2 <= -2.8e-129) {
		tmp = t_0;
	} else if (phi2 <= 4.5e-305) {
		tmp = R * lambda2;
	} else if (phi2 <= 1.8e-279) {
		tmp = t_0;
	} else if (phi2 <= 9.8e+70) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = phi1 * -R
	tmp = 0
	if phi2 <= -2.8e-129:
		tmp = t_0
	elif phi2 <= 4.5e-305:
		tmp = R * lambda2
	elif phi2 <= 1.8e-279:
		tmp = t_0
	elif phi2 <= 9.8e+70:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(phi1 * Float64(-R))
	tmp = 0.0
	if (phi2 <= -2.8e-129)
		tmp = t_0;
	elseif (phi2 <= 4.5e-305)
		tmp = Float64(R * lambda2);
	elseif (phi2 <= 1.8e-279)
		tmp = t_0;
	elseif (phi2 <= 9.8e+70)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = phi1 * -R;
	tmp = 0.0;
	if (phi2 <= -2.8e-129)
		tmp = t_0;
	elseif (phi2 <= 4.5e-305)
		tmp = R * lambda2;
	elseif (phi2 <= 1.8e-279)
		tmp = t_0;
	elseif (phi2 <= 9.8e+70)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(phi1 * (-R)), $MachinePrecision]}, If[LessEqual[phi2, -2.8e-129], t$95$0, If[LessEqual[phi2, 4.5e-305], N[(R * lambda2), $MachinePrecision], If[LessEqual[phi2, 1.8e-279], t$95$0, If[LessEqual[phi2, 9.8e+70], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.7999999999999999e-129 or 4.5000000000000002e-305 < phi2 < 1.7999999999999998e-279

   1. Initial program 58.8%

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

      1. hypot-define 94.2%

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

   3. Simplified 94.2%

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

   5. Taylor expanded in phi1 around -inf 15.1%

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

      1. mul-1-neg 15.1%

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

   7. Simplified 15.1%

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

   if -2.7999999999999999e-129 < phi2 < 4.5000000000000002e-305 or 1.7999999999999998e-279 < phi2 < 9.80000000000000056e70

   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 97.2%

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

   3. Simplified 97.2%

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

   5. Taylor expanded in phi2 around 0 94.2%

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

   6. Taylor expanded in phi1 around 0 87.9%

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

   7. Taylor expanded in lambda2 around inf 24.2%

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

   if 9.80000000000000056e70 < phi2

   1. Initial program 50.6%

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

      1. hypot-define 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in phi2 around inf 73.6%

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

      1. *-commutative 73.6%

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

   7. Simplified 73.6%

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

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.8 \cdot 10^{-129}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 4.5 \cdot 10^{-305}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{elif}\;\phi_2 \leq 1.8 \cdot 10^{-279}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 9.8 \cdot 10^{+70}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.1% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7.2 \cdot 10^{-118}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 6800000:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 -7.2e-118)
   (* phi1 (- R))
   (if (<= phi2 6800000.0)
     (* R (* lambda2 (- 1.0 (/ 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 <= -7.2e-118) {
		tmp = phi1 * -R;
	} else if (phi2 <= 6800000.0) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} 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 <= (-7.2d-118)) then
        tmp = phi1 * -r
    else if (phi2 <= 6800000.0d0) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / lambda2)))
    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 <= -7.2e-118) {
		tmp = phi1 * -R;
	} else if (phi2 <= 6800000.0) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= -7.2e-118:
		tmp = phi1 * -R
	elif phi2 <= 6800000.0:
		tmp = R * (lambda2 * (1.0 - (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 <= -7.2e-118)
		tmp = Float64(phi1 * Float64(-R));
	elseif (phi2 <= 6800000.0)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - 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 <= -7.2e-118)
		tmp = phi1 * -R;
	elseif (phi2 <= 6800000.0)
		tmp = R * (lambda2 * (1.0 - (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, -7.2e-118], N[(phi1 * (-R)), $MachinePrecision], If[LessEqual[phi2, 6800000.0], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -7.2000000000000004e-118

   1. Initial program 56.0%

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

      1. hypot-define 93.4%

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

   3. Simplified 93.4%

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

   5. Taylor expanded in phi1 around -inf 12.5%

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

      1. mul-1-neg 12.5%

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

   7. Simplified 12.5%

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

   if -7.2000000000000004e-118 < phi2 < 6.8e6

   1. Initial program 64.4%

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

      1. hypot-define 99.2%

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

   3. Simplified 99.2%

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

   5. Taylor expanded in phi2 around 0 97.6%

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

   6. Taylor expanded in phi1 around 0 87.9%

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

   7. Taylor expanded in lambda2 around inf 30.5%

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

      1. associate-*r/ 30.5%

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

      2. mul-1-neg 30.5%

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

   9. Simplified 30.5%

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

   if 6.8e6 < phi2

   1. Initial program 51.5%

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

      1. hypot-define 88.7%

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

   3. Simplified 88.7%

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

   5. Taylor expanded in phi2 around inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

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

   7. Simplified 60.9%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -7.2 \cdot 10^{-118}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 6800000:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 29.4% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.15 \cdot 10^{+169}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -2.15e+169)
   (* R (- lambda1))
   (if (<= lambda1 6e-37) (* phi2 (- R (* phi1 (/ R phi2)))) (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.15e+169) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 6e-37) {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda1 <= (-2.15d+169)) then
        tmp = r * -lambda1
    else if (lambda1 <= 6d-37) then
        tmp = phi2 * (r - (phi1 * (r / phi2)))
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -2.15e+169) {
		tmp = R * -lambda1;
	} else if (lambda1 <= 6e-37) {
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.15e+169:
		tmp = R * -lambda1
	elif lambda1 <= 6e-37:
		tmp = phi2 * (R - (phi1 * (R / phi2)))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -2.15e+169)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda1 <= 6e-37)
		tmp = Float64(phi2 * Float64(R - Float64(phi1 * Float64(R / phi2))));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -2.15e+169)
		tmp = R * -lambda1;
	elseif (lambda1 <= 6e-37)
		tmp = phi2 * (R - (phi1 * (R / phi2)));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -2.15e+169], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda1, 6e-37], N[(phi2 * N[(R - N[(phi1 * N[(R / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -2.1500000000000001e169

   1. Initial program 45.2%

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

      1. hypot-define 93.8%

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

   3. Simplified 93.8%

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

   5. Taylor expanded in phi2 around 0 84.9%

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

   6. Taylor expanded in phi1 around 0 82.0%

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

   7. Taylor expanded in lambda1 around -inf 67.8%

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

      1. associate-*r* 67.8%

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

      2. neg-mul-1 67.8%

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

   9. Simplified 67.8%

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

   if -2.1500000000000001e169 < lambda1 < 6e-37

   1. Initial program 64.6%

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

      1. hypot-define 96.2%

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

   3. Simplified 96.2%

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

   5. Taylor expanded in phi2 around inf 25.6%

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

      1. mul-1-neg 25.6%

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

      2. unsub-neg 25.6%

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

      3. *-commutative 25.6%

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

      4. associate-/l* 29.0%

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

   7. Simplified 29.0%

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

   if 6e-37 < lambda1

   1. Initial program 53.7%

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

      1. hypot-define 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in phi2 around 0 85.4%

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

   6. Taylor expanded in phi1 around 0 81.3%

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

   7. Taylor expanded in lambda2 around inf 9.3%

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

4. Final simplification 25.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.15 \cdot 10^{+169}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_1 \leq 6 \cdot 10^{-37}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.9% accurate, 17.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{-225}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 -8.5e-225)
   (* R (- lambda1))
   (if (<= lambda2 1.12e+117)
     (* R (* phi2 (- 1.0 (/ phi1 phi2))))
     (* R lambda2))))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -8.5e-225) {
		tmp = R * -lambda1;
	} else if (lambda2 <= 1.12e+117) {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= (-8.5d-225)) then
        tmp = r * -lambda1
    else if (lambda2 <= 1.12d+117) then
        tmp = r * (phi2 * (1.0d0 - (phi1 / phi2)))
    else
        tmp = r * lambda2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= -8.5e-225) {
		tmp = R * -lambda1;
	} else if (lambda2 <= 1.12e+117) {
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	} else {
		tmp = R * lambda2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= -8.5e-225:
		tmp = R * -lambda1
	elif lambda2 <= 1.12e+117:
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)))
	else:
		tmp = R * lambda2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= -8.5e-225)
		tmp = Float64(R * Float64(-lambda1));
	elseif (lambda2 <= 1.12e+117)
		tmp = Float64(R * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	else
		tmp = Float64(R * lambda2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= -8.5e-225)
		tmp = R * -lambda1;
	elseif (lambda2 <= 1.12e+117)
		tmp = R * (phi2 * (1.0 - (phi1 / phi2)));
	else
		tmp = R * lambda2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, -8.5e-225], N[(R * (-lambda1)), $MachinePrecision], If[LessEqual[lambda2, 1.12e+117], N[(R * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * lambda2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda2 < -8.4999999999999998e-225

   1. Initial program 53.6%

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

      1. hypot-define 95.4%

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

   3. Simplified 95.4%

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

   5. Taylor expanded in phi2 around 0 87.3%

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

   6. Taylor expanded in phi1 around 0 81.3%

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

   7. Taylor expanded in lambda1 around -inf 17.0%

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

      1. associate-*r* 17.0%

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

      2. neg-mul-1 17.0%

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

   9. Simplified 17.0%

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

   if -8.4999999999999998e-225 < lambda2 < 1.12000000000000002e117

   1. Initial program 67.5%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi2 around inf 29.0%

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

      1. mul-1-neg 29.0%

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

      2. unsub-neg 29.0%

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

   7. Simplified 29.0%

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

   if 1.12000000000000002e117 < lambda2

   1. Initial program 51.5%

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

      1. hypot-define 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in phi2 around 0 81.5%

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

   6. Taylor expanded in phi1 around 0 72.4%

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

   7. Taylor expanded in lambda2 around inf 59.5%

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

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq -8.5 \cdot 10^{-225}:\\ \;\;\;\;R \cdot \left(-\lambda_1\right)\\ \mathbf{elif}\;\lambda_2 \leq 1.12 \cdot 10^{+117}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \lambda_2\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{+69}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 3.2e+69) (* R lambda2) (* R phi2)))

C

double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.2e+69) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 3.2d+69) then
        tmp = r * lambda2
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3.2e+69) {
		tmp = R * lambda2;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 3.2e+69:
		tmp = R * lambda2
	else:
		tmp = R * phi2
	return tmp

Julia

function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 3.2e+69)
		tmp = Float64(R * lambda2);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 3.2e+69)
		tmp = R * lambda2;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 3.2e+69], N[(R * lambda2), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.19999999999999985e69

   1. Initial program 60.3%

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

      1. hypot-define 95.7%

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

   3. Simplified 95.7%

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

   5. Taylor expanded in phi2 around 0 89.0%

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

   6. Taylor expanded in phi1 around 0 83.0%

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

   7. Taylor expanded in lambda2 around inf 17.7%

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

   if 3.19999999999999985e69 < phi2

   1. Initial program 50.6%

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

      1. hypot-define 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in phi2 around inf 73.6%

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

      1. *-commutative 73.6%

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

   7. Simplified 73.6%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.2 \cdot 10^{+69}:\\ \;\;\;\;R \cdot \lambda_2\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 14.1% accurate, 109.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 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 95.0%

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

3. Simplified 95.0%

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

5. Taylor expanded in phi2 around 0 88.3%

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

6. Taylor expanded in phi1 around 0 83.3%

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

7. Taylor expanded in lambda2 around inf 16.2%

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

Reproduce

herbie shell --seed 2024114 
(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))))))