Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.0% → 88.8%

Time: 10.4s

Alternatives: 14

Speedup: 2.4×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.0% 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: 88.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5e+60)
   (* (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1) R)
   (* (hypot (* (cos (* phi2 0.5)) (- lambda1 lambda2)) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5e+60) {
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5e+60) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5e+60:
		tmp = math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R
	else:
		tmp = math.hypot((math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5e+60)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(phi2 * 0.5)) * Float64(lambda1 - lambda2)), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5e+60)
		tmp = hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1) * R;
	else
		tmp = hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5e+60], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.99999999999999975e60

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   if -4.99999999999999975e60 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 79.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9.8e+61)
   (* (hypot (* (cos (* 0.5 phi1)) lambda1) phi1) R)
   (* (hypot (* (cos (* phi2 0.5)) (- lambda1 lambda2)) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.8e+61) {
		tmp = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	} else {
		tmp = hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.8e+61) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * lambda1), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9.8e+61:
		tmp = math.hypot((math.cos((0.5 * phi1)) * lambda1), phi1) * R
	else:
		tmp = math.hypot((math.cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9.8e+61)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(phi2 * 0.5)) * Float64(lambda1 - lambda2)), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9.8e+61)
		tmp = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	else
		tmp = hypot((cos((phi2 * 0.5)) * (lambda1 - lambda2)), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9.8e+61], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.8000000000000005e61

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in lambda1 around inf

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   if -9.8000000000000005e61 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9.8e+61)
   (* (hypot (* (cos (* 0.5 phi1)) lambda1) phi1) R)
   (* (hypot (- lambda1 lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.8e+61) {
		tmp = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	} else {
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9.8e+61) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * lambda1), phi1) * R;
	} else {
		tmp = Math.hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9.8e+61:
		tmp = math.hypot((math.cos((0.5 * phi1)) * lambda1), phi1) * R
	else:
		tmp = math.hypot((lambda1 - lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9.8e+61)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), phi1) * R);
	else
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9.8e+61)
		tmp = hypot((cos((0.5 * phi1)) * lambda1), phi1) * R;
	else
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9.8e+61], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.8000000000000005e61

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in lambda1 around inf

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right), \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 79.1%

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

   if -9.8000000000000005e61 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9.8 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.4 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.4e+60)
   (* (hypot (* (cos (* 0.5 phi1)) lambda2) phi1) R)
   (* (hypot (- lambda1 lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.4e+60) {
		tmp = hypot((cos((0.5 * phi1)) * lambda2), phi1) * R;
	} else {
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.4e+60) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * lambda2), phi1) * R;
	} else {
		tmp = Math.hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.4e+60:
		tmp = math.hypot((math.cos((0.5 * phi1)) * lambda2), phi1) * R
	else:
		tmp = math.hypot((lambda1 - lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.4e+60)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda2), phi1) * R);
	else
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.4e+60)
		tmp = hypot((cos((0.5 * phi1)) * lambda2), phi1) * R;
	else
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.4e+60], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.39999999999999982e60

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} + {\phi_1}^{2}} \]
   7. Step-by-step derivation

   8. Applied rewrites 83.3%

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

   if -6.39999999999999982e60 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.9%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.4 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9e+60)
   (* (hypot (- lambda1 lambda2) phi1) R)
   (* (hypot (- lambda1 lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9e+60) {
		tmp = hypot((lambda1 - lambda2), phi1) * R;
	} else {
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9e+60) {
		tmp = Math.hypot((lambda1 - lambda2), phi1) * R;
	} else {
		tmp = Math.hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9e+60:
		tmp = math.hypot((lambda1 - lambda2), phi1) * R
	else:
		tmp = math.hypot((lambda1 - lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9e+60)
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi1) * R);
	else
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9e+60)
		tmp = hypot((lambda1 - lambda2), phi1) * R;
	else
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9e+60], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.00000000000000026e60

   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. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 86.2

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 81.6%

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

   if -9.00000000000000026e60 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.9%

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

4. Final simplification 74.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\phi_1} - 1\right) \cdot \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -7.2e+61)
   (* (* (- (/ phi2 phi1) 1.0) phi1) R)
   (* (hypot (- lambda1 lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.2e+61) {
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	} else {
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.2e+61) {
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	} else {
		tmp = Math.hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -7.2e+61:
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R
	else:
		tmp = math.hypot((lambda1 - lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -7.2e+61)
		tmp = Float64(Float64(Float64(Float64(phi2 / phi1) - 1.0) * phi1) * R);
	else
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -7.2e+61)
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	else
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -7.2e+61], N[(N[(N[(N[(phi2 / phi1), $MachinePrecision] - 1.0), $MachinePrecision] * phi1), $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.20000000000000021e61

   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. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -7.20000000000000021e61 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.9%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.2 \cdot 10^{+61}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\phi_1} - 1\right) \cdot \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 67.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\phi_1} - 1\right) \cdot \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(-\lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -5.5e+60)
   (* (* (- (/ phi2 phi1) 1.0) phi1) R)
   (* (hypot (- lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e+60) {
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	} else {
		tmp = hypot(-lambda2, phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -5.5e+60) {
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	} else {
		tmp = Math.hypot(-lambda2, phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -5.5e+60:
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R
	else:
		tmp = math.hypot(-lambda2, phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -5.5e+60)
		tmp = Float64(Float64(Float64(Float64(phi2 / phi1) - 1.0) * phi1) * R);
	else
		tmp = Float64(hypot(Float64(-lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -5.5e+60)
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	else
		tmp = hypot(-lambda2, phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -5.5e+60], N[(N[(N[(N[(phi2 / phi1), $MachinePrecision] - 1.0), $MachinePrecision] * phi1), $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[(-lambda2) ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.5000000000000001e60

   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. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 77.5

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

   5. Applied rewrites 77.5%

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

   if -5.5000000000000001e60 < phi1

   1. Initial program 62.4%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 72.9%

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

   9. Taylor expanded in lambda1 around 0

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

   11. Applied rewrites 56.2%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -5.5 \cdot 10^{+60}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\phi_1} - 1\right) \cdot \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(-\lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{+124}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\left(R - \frac{R \cdot \phi_1}{\phi_2}\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.2e+124)
   (* (- phi1) R)
   (if (<= phi1 -1.36e+16)
     (* (- R (/ (* R phi1) phi2)) phi2)
     (if (<= phi1 9e-290) (* (- lambda1) R) (* phi2 R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e+124) {
		tmp = -phi1 * R;
	} else if (phi1 <= -1.36e+16) {
		tmp = (R - ((R * phi1) / phi2)) * phi2;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-4.2d+124)) then
        tmp = -phi1 * r
    else if (phi1 <= (-1.36d+16)) then
        tmp = (r - ((r * phi1) / phi2)) * phi2
    else if (phi1 <= 9d-290) then
        tmp = -lambda1 * r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.2e+124) {
		tmp = -phi1 * R;
	} else if (phi1 <= -1.36e+16) {
		tmp = (R - ((R * phi1) / phi2)) * phi2;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.2e+124:
		tmp = -phi1 * R
	elif phi1 <= -1.36e+16:
		tmp = (R - ((R * phi1) / phi2)) * phi2
	elif phi1 <= 9e-290:
		tmp = -lambda1 * R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.2e+124)
		tmp = Float64(Float64(-phi1) * R);
	elseif (phi1 <= -1.36e+16)
		tmp = Float64(Float64(R - Float64(Float64(R * phi1) / phi2)) * phi2);
	elseif (phi1 <= 9e-290)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.2e+124)
		tmp = -phi1 * R;
	elseif (phi1 <= -1.36e+16)
		tmp = (R - ((R * phi1) / phi2)) * phi2;
	elseif (phi1 <= 9e-290)
		tmp = -lambda1 * R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.2e+124], N[((-phi1) * R), $MachinePrecision], If[LessEqual[phi1, -1.36e+16], N[(N[(R - N[(N[(R * phi1), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi1, 9e-290], N[((-lambda1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -4.20000000000000023e124

   1. Initial program 48.0%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.6

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

   5. Applied rewrites 82.6%

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

   if -4.20000000000000023e124 < phi1 < -1.36e16

   1. Initial program 64.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_1\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_1}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

   6. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 67.5

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

   8. Applied rewrites 67.5%

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

   if -1.36e16 < phi1 < 9e-290

   1. Initial program 66.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot \frac{1}{2}\right)\right) \]

      9. lower-+.f64 22.2

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

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\lambda_1 \cdot \left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.3%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 9e-290 < phi1

   1. Initial program 60.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 20.1

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

   5. Applied rewrites 20.1%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.2 \cdot 10^{+124}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\left(R - \frac{R \cdot \phi_1}{\phi_2}\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 49.3% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{+89}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\left(R - \frac{\phi_1}{\phi_2} \cdot R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4.5e+89)
   (* (- phi1) R)
   (if (<= phi1 -1.36e+16)
     (* (- R (* (/ phi1 phi2) R)) phi2)
     (if (<= phi1 9e-290) (* (- lambda1) R) (* phi2 R)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.5e+89) {
		tmp = -phi1 * R;
	} else if (phi1 <= -1.36e+16) {
		tmp = (R - ((phi1 / phi2) * R)) * phi2;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-4.5d+89)) then
        tmp = -phi1 * r
    else if (phi1 <= (-1.36d+16)) then
        tmp = (r - ((phi1 / phi2) * r)) * phi2
    else if (phi1 <= 9d-290) then
        tmp = -lambda1 * r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4.5e+89) {
		tmp = -phi1 * R;
	} else if (phi1 <= -1.36e+16) {
		tmp = (R - ((phi1 / phi2) * R)) * phi2;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4.5e+89:
		tmp = -phi1 * R
	elif phi1 <= -1.36e+16:
		tmp = (R - ((phi1 / phi2) * R)) * phi2
	elif phi1 <= 9e-290:
		tmp = -lambda1 * R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4.5e+89)
		tmp = Float64(Float64(-phi1) * R);
	elseif (phi1 <= -1.36e+16)
		tmp = Float64(Float64(R - Float64(Float64(phi1 / phi2) * R)) * phi2);
	elseif (phi1 <= 9e-290)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4.5e+89)
		tmp = -phi1 * R;
	elseif (phi1 <= -1.36e+16)
		tmp = (R - ((phi1 / phi2) * R)) * phi2;
	elseif (phi1 <= 9e-290)
		tmp = -lambda1 * R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4.5e+89], N[((-phi1) * R), $MachinePrecision], If[LessEqual[phi1, -1.36e+16], N[(N[(R - N[(N[(phi1 / phi2), $MachinePrecision] * R), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi1, 9e-290], N[((-lambda1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -4.5e89

   1. Initial program 49.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if -4.5e89 < phi1 < -1.36e16

   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. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 69.3

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

   5. Applied rewrites 69.3%

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

   if -1.36e16 < phi1 < 9e-290

   1. Initial program 66.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot \frac{1}{2}\right)\right) \]

      9. lower-+.f64 22.2

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

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\lambda_1 \cdot \left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.3%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 9e-290 < phi1

   1. Initial program 60.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 20.1

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

   5. Applied rewrites 20.1%

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

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.5 \cdot 10^{+89}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\left(R - \frac{\phi_1}{\phi_2} \cdot R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_2}{\phi_1}, R\right) \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.36e+16)
   (* (fma R (/ (- phi2) phi1) R) (- phi1))
   (if (<= phi1 9e-290) (* (- lambda1) R) (* phi2 R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.36e+16) {
		tmp = fma(R, (-phi2 / phi1), R) * -phi1;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.36e+16)
		tmp = Float64(fma(R, Float64(Float64(-phi2) / phi1), R) * Float64(-phi1));
	elseif (phi1 <= 9e-290)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.36e+16], N[(N[(R * N[((-phi2) / phi1), $MachinePrecision] + R), $MachinePrecision] * (-phi1)), $MachinePrecision], If[LessEqual[phi1, 9e-290], N[((-lambda1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.36e16

   1. Initial program 54.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. associate-/l* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. associate-*r/ N/A

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

      12. lower-/.f64 N/A

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

      13. mul-1-neg N/A

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

      14. lower-neg.f64 73.6

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

   5. Applied rewrites 73.6%

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

   if -1.36e16 < phi1 < 9e-290

   1. Initial program 66.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot \frac{1}{2}\right)\right) \]

      9. lower-+.f64 22.2

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

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\lambda_1 \cdot \left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.3%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 9e-290 < phi1

   1. Initial program 60.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 20.1

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

   5. Applied rewrites 20.1%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_2}{\phi_1}, R\right) \cdot \left(-\phi_1\right)\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.3% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\phi_1} - 1\right) \cdot \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.36e+16)
   (* (* (- (/ phi2 phi1) 1.0) phi1) R)
   (if (<= phi1 9e-290) (* (- lambda1) R) (* phi2 R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.36e+16) {
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-1.36d+16)) then
        tmp = (((phi2 / phi1) - 1.0d0) * phi1) * r
    else if (phi1 <= 9d-290) then
        tmp = -lambda1 * r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.36e+16) {
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	} else if (phi1 <= 9e-290) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.36e+16:
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R
	elif phi1 <= 9e-290:
		tmp = -lambda1 * R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.36e+16)
		tmp = Float64(Float64(Float64(Float64(phi2 / phi1) - 1.0) * phi1) * R);
	elseif (phi1 <= 9e-290)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.36e+16)
		tmp = (((phi2 / phi1) - 1.0) * phi1) * R;
	elseif (phi1 <= 9e-290)
		tmp = -lambda1 * R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.36e+16], N[(N[(N[(N[(phi2 / phi1), $MachinePrecision] - 1.0), $MachinePrecision] * phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, 9e-290], N[((-lambda1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.36e16

   1. Initial program 54.3%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 73.7

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

   5. Applied rewrites 73.7%

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

   if -1.36e16 < phi1 < 9e-290

   1. Initial program 66.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot \frac{1}{2}\right)\right) \]

      9. lower-+.f64 22.2

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

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\lambda_1 \cdot \left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.3%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 9e-290 < phi1

   1. Initial program 60.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 20.1

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

   5. Applied rewrites 20.1%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.36 \cdot 10^{+16}:\\ \;\;\;\;\left(\left(\frac{\phi_2}{\phi_1} - 1\right) \cdot \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 9 \cdot 10^{-290}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 47.3% accurate, 10.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.45 \cdot 10^{+16} \lor \neg \left(\phi_1 \leq 9 \cdot 10^{-290}\right):\\ \;\;\;\;\phi_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -9e+60)
   (* (- phi1) R)
   (if (or (<= phi1 -1.45e+16) (not (<= phi1 9e-290)))
     (* phi2 R)
     (* (- lambda1) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9e+60) {
		tmp = -phi1 * R;
	} else if ((phi1 <= -1.45e+16) || !(phi1 <= 9e-290)) {
		tmp = phi2 * R;
	} else {
		tmp = -lambda1 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-9d+60)) then
        tmp = -phi1 * r
    else if ((phi1 <= (-1.45d+16)) .or. (.not. (phi1 <= 9d-290))) then
        tmp = phi2 * r
    else
        tmp = -lambda1 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -9e+60) {
		tmp = -phi1 * R;
	} else if ((phi1 <= -1.45e+16) || !(phi1 <= 9e-290)) {
		tmp = phi2 * R;
	} else {
		tmp = -lambda1 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -9e+60:
		tmp = -phi1 * R
	elif (phi1 <= -1.45e+16) or not (phi1 <= 9e-290):
		tmp = phi2 * R
	else:
		tmp = -lambda1 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -9e+60)
		tmp = Float64(Float64(-phi1) * R);
	elseif ((phi1 <= -1.45e+16) || !(phi1 <= 9e-290))
		tmp = Float64(phi2 * R);
	else
		tmp = Float64(Float64(-lambda1) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -9e+60)
		tmp = -phi1 * R;
	elseif ((phi1 <= -1.45e+16) || ~((phi1 <= 9e-290)))
		tmp = phi2 * R;
	else
		tmp = -lambda1 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -9e+60], N[((-phi1) * R), $MachinePrecision], If[Or[LessEqual[phi1, -1.45e+16], N[Not[LessEqual[phi1, 9e-290]], $MachinePrecision]], N[(phi2 * R), $MachinePrecision], N[((-lambda1) * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -9.00000000000000026e60

   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. Add Preprocessing

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -9.00000000000000026e60 < phi1 < -1.45e16 or 9e-290 < phi1

   1. Initial program 59.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 23.1

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

   5. Applied rewrites 23.1%

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

   if -1.45e16 < phi1 < 9e-290

   1. Initial program 66.6%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot \frac{1}{2}\right)\right) \]

      9. lower-+.f64 22.2

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

   5. Applied rewrites 22.2%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\lambda_1 \cdot \left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 21.2%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 19.3%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -9 \cdot 10^{+60}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.45 \cdot 10^{+16} \lor \neg \left(\phi_1 \leq 9 \cdot 10^{-290}\right):\\ \;\;\;\;\phi_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 37.6% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.26 \cdot 10^{-20}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.26e-20) (* (- lambda1) R) (* phi2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.26e-20) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.26d-20) then
        tmp = -lambda1 * r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.26e-20) {
		tmp = -lambda1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.26e-20:
		tmp = -lambda1 * R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.26e-20)
		tmp = Float64(Float64(-lambda1) * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.26e-20)
		tmp = -lambda1 * R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.26e-20], N[((-lambda1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.26e-20

   1. Initial program 61.2%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. Add Preprocessing

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(\left(-1 \cdot \lambda_1\right) \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right)} \]

      3. mul-1-neg N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(\mathsf{neg}\left(\lambda_1\right)\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto R \cdot \left(\color{blue}{\left(-\lambda_1\right)} \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)\right) \]

      5. lower-cos.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \color{blue}{\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}\right) \]

      6. *-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      7. lower-*.f64 N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \color{blue}{\left(\left(\phi_1 + \phi_2\right) \cdot \frac{1}{2}\right)}\right) \]

      8. +-commutative N/A

         \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \cos \left(\color{blue}{\left(\phi_2 + \phi_1\right)} \cdot \frac{1}{2}\right)\right) \]

      9. lower-+.f64 15.7

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

   5. Applied rewrites 15.7%

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

   6. Taylor expanded in phi2 around 0

      \[\leadsto R \cdot \left(-1 \cdot \left(\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \phi_1\right)\right) + \color{blue}{\frac{1}{2} \cdot \left(\lambda_1 \cdot \left(\phi_2 \cdot \sin \left(\frac{1}{2} \cdot \phi_1\right)\right)\right)}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 16.7%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(-1 \cdot \lambda_1\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 14.9%

       \[\leadsto R \cdot \left(-\lambda_1\right) \]

   if 1.26e-20 < phi2

   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. Add Preprocessing

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.26 \cdot 10^{-20}:\\ \;\;\;\;\left(-\lambda_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 31.0% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \phi_2 \cdot R \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2) :precision binary64 (* phi2 R))

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 = phi2 * r
end function

Java

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

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	return phi2 * R

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	return Float64(phi2 * R)
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = phi2 * R;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(phi2 * R), $MachinePrecision]

Derivation

1. Initial program 60.7%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
2. Add Preprocessing

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 18.2

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

5. Applied rewrites 18.2%

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

6. Final simplification 18.2%

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

Reproduce

herbie shell --seed 2024271 
(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))))))