Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.7% → 90.5%

Time: 13.9s

Alternatives: 13

Speedup: 18.6×

• 31.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.7% 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: 90.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 2.05e-44)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (* R (hypot phi2 (* (- lambda1 lambda2) (cos (* phi2 0.5)))))))

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 <= 2.05e-44) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	}
	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 (phi2 <= 2.05e-44) {
		tmp = Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * Math.hypot(phi2, ((lambda1 - lambda2) * Math.cos((phi2 * 0.5))));
	}
	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 <= 2.05e-44:
		tmp = math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5)))) * R
	else:
		tmp = R * math.hypot(phi2, ((lambda1 - lambda2) * math.cos((phi2 * 0.5))))
	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 <= 2.05e-44)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	else
		tmp = Float64(R * hypot(phi2, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5)))));
	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 <= 2.05e-44)
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	else
		tmp = R * hypot(phi2, ((lambda1 - lambda2) * cos((phi2 * 0.5))));
	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, 2.05e-44], N[(N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.04999999999999996e-44

   1. Initial program 60.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if 2.04999999999999996e-44 < phi2

   1. Initial program 54.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 75.5

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

   5. Applied rewrites 75.5%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.05 \cdot 10^{-44}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 7.2e+42)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (* R (hypot phi2 (* lambda1 (cos (* phi2 0.5)))))))

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 <= 7.2e+42) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * hypot(phi2, (lambda1 * cos((phi2 * 0.5))));
	}
	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 (phi2 <= 7.2e+42) {
		tmp = Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 * Math.cos((phi2 * 0.5))));
	}
	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 <= 7.2e+42:
		tmp = math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5)))) * R
	else:
		tmp = R * math.hypot(phi2, (lambda1 * math.cos((phi2 * 0.5))))
	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 <= 7.2e+42)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 * cos(Float64(phi2 * 0.5)))));
	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 <= 7.2e+42)
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	else
		tmp = R * hypot(phi2, (lambda1 * cos((phi2 * 0.5))));
	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, 7.2e+42], N[(N[Sqrt[phi1 ^ 2 + N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 7.2000000000000002e42

   1. Initial program 60.1%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if 7.2000000000000002e42 < phi2

   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 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 64.1

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

   8. Applied rewrites 64.1%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7.2 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.9 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 -4.9e+47)
   (* R (hypot phi1 (* lambda1 (cos (* phi1 0.5)))))
   (* R (hypot phi2 (- lambda1 lambda2)))))

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 <= -4.9e+47) {
		tmp = R * hypot(phi1, (lambda1 * cos((phi1 * 0.5))));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	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 <= -4.9e+47) {
		tmp = R * Math.hypot(phi1, (lambda1 * Math.cos((phi1 * 0.5))));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	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 <= -4.9e+47:
		tmp = R * math.hypot(phi1, (lambda1 * math.cos((phi1 * 0.5))))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	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 <= -4.9e+47)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 * cos(Float64(phi1 * 0.5)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	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 <= -4.9e+47)
		tmp = R * hypot(phi1, (lambda1 * cos((phi1 * 0.5))));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	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, -4.9e+47], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.9000000000000003e47

   1. Initial program 47.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 72.3

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

   5. Applied rewrites 72.3%

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

   6. Taylor expanded in lambda2 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. lower-hypot.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. lower-*.f64 65.9

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

   8. Applied rewrites 65.9%

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

   if -4.9000000000000003e47 < phi1

   1. Initial program 62.6%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 76.0

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

   5. Applied rewrites 76.0%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower--.f64 67.6

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

   8. Applied rewrites 67.6%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4.9 \cdot 10^{+47}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{+123}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), 0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

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 -1.2e+123)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi1 -4.8e+18)
     (*
      R
      (sqrt
       (fma
        (* (- lambda1 lambda2) (- lambda1 lambda2))
        (+ 0.5 (* 0.5 (cos (+ phi2 phi1))))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* R (hypot phi2 (- lambda1 lambda2))))))

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 <= -1.2e+123) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi1 <= -4.8e+18) {
		tmp = R * sqrt(fma(((lambda1 - lambda2) * (lambda1 - lambda2)), (0.5 + (0.5 * cos((phi2 + phi1)))), ((phi1 - phi2) * (phi1 - phi2))));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	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 <= -1.2e+123)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi1 <= -4.8e+18)
		tmp = Float64(R * sqrt(fma(Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)), Float64(0.5 + Float64(0.5 * cos(Float64(phi2 + phi1)))), Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	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, -1.2e+123], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi1, -4.8e+18], N[(R * N[Sqrt[N[(N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] * N[(0.5 + N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.19999999999999994e123

   1. Initial program 49.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 80.1

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

   8. Applied rewrites 80.1%

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

   if -1.19999999999999994e123 < phi1 < -4.8e18

   1. Initial program 49.9%

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

   4. Applied rewrites 49.9%

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

   if -4.8e18 < phi1

   1. Initial program 62.5%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 76.6

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

   5. Applied rewrites 76.6%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower--.f64 67.8

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

   8. Applied rewrites 67.8%

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

4. Final simplification 68.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.2 \cdot 10^{+123}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_1 \leq -4.8 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), 0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{+98}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

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 -3.6e+98)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (hypot phi2 (- lambda1 lambda2)))))

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 <= -3.6e+98) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	}
	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 <= -3.6e+98) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * Math.hypot(phi2, (lambda1 - lambda2));
	}
	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 <= -3.6e+98:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * math.hypot(phi2, (lambda1 - lambda2))
	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 <= -3.6e+98)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * hypot(phi2, Float64(lambda1 - lambda2)));
	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 <= -3.6e+98)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * hypot(phi2, (lambda1 - lambda2));
	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, -3.6e+98], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.59999999999999981e98

   1. Initial program 49.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 80.9

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 73.7

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

   8. Applied rewrites 73.7%

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

   if -3.59999999999999981e98 < phi1

   1. Initial program 61.3%

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

   3. Taylor expanded in phi1 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-cos.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 74.3

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

   5. Applied rewrites 74.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. lower--.f64 66.4

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

   8. Applied rewrites 66.4%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.6 \cdot 10^{+98}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 9.5e+45)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* R (- phi2 phi1))))

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 <= 9.5e+45) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 (phi2 <= 9.5e+45) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= 9.5e+45:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R * (phi2 - phi1)
	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 <= 9.5e+45)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	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 <= 9.5e+45)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R * (phi2 - phi1);
	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, 9.5e+45], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 9.4999999999999998e45

   1. Initial program 60.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 68.6

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

   8. Applied rewrites 68.6%

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

   if 9.4999999999999998e45 < phi2

   1. Initial program 53.3%

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

   3. Taylor expanded in phi2 around 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 55.1

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

   8. Applied rewrites 55.1%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 4.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.32 \cdot 10^{-194}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-122}:\\ \;\;\;\;R \cdot \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_1 \cdot \phi_1\right)}}}\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{+45}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{\lambda_1 \cdot R}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 -1.32e-194)
   (* phi1 (- R))
   (if (<= phi2 2.2e-122)
     (*
      R
      (/
       1.0
       (sqrt
        (/ 1.0 (fma (- lambda1 lambda2) (- lambda1 lambda2) (* phi1 phi1))))))
     (if (<= phi2 1.15e+45)
       (* lambda2 (- R (/ (* lambda1 R) lambda2)))
       (* R (- phi2 phi1))))))

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 <= -1.32e-194) {
		tmp = phi1 * -R;
	} else if (phi2 <= 2.2e-122) {
		tmp = R * (1.0 / sqrt((1.0 / fma((lambda1 - lambda2), (lambda1 - lambda2), (phi1 * phi1)))));
	} else if (phi2 <= 1.15e+45) {
		tmp = lambda2 * (R - ((lambda1 * R) / lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= -1.32e-194)
		tmp = Float64(phi1 * Float64(-R));
	elseif (phi2 <= 2.2e-122)
		tmp = Float64(R * Float64(1.0 / sqrt(Float64(1.0 / fma(Float64(lambda1 - lambda2), Float64(lambda1 - lambda2), Float64(phi1 * phi1))))));
	elseif (phi2 <= 1.15e+45)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(lambda1 * R) / lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	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[phi2, -1.32e-194], N[(phi1 * (-R)), $MachinePrecision], If[LessEqual[phi2, 2.2e-122], N[(R * N[(1.0 / N[Sqrt[N[(1.0 / N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision] + N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.15e+45], N[(lambda2 * N[(R - N[(N[(lambda1 * R), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < -1.32e-194

   1. Initial program 58.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 16.3

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

   5. Applied rewrites 16.3%

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

   if -1.32e-194 < phi2 < 2.2e-122

   1. Initial program 64.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 79.2

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

   8. Applied rewrites 79.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. flip3-+ N/A

         \[\leadsto \sqrt{\color{blue}{\frac{{\left(\phi_1 \cdot \phi_1\right)}^{3} + {\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)}^{3}}{\left(\phi_1 \cdot \phi_1\right) \cdot \left(\phi_1 \cdot \phi_1\right) + \left(\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) - \left(\phi_1 \cdot \phi_1\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)}}} \cdot R \]

      6. clear-num N/A

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

      7. sqrt-div N/A

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

      8. metadata-eval N/A

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

   10. Applied rewrites 59.1%

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

   if 2.2e-122 < phi2 < 1.15000000000000006e45

   1. Initial program 58.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 82.9

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

   5. Applied rewrites 82.9%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 82.9

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

   8. Applied rewrites 82.9%

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

   9. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 22.4

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

   11. Applied rewrites 22.4%

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

   if 1.15000000000000006e45 < phi2

   1. Initial program 53.3%

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

   3. Taylor expanded in phi2 around 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 55.1

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

   8. Applied rewrites 55.1%

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

4. Final simplification 34.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq -1.32 \cdot 10^{-194}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-122}:\\ \;\;\;\;R \cdot \frac{1}{\sqrt{\frac{1}{\mathsf{fma}\left(\lambda_1 - \lambda_2, \lambda_1 - \lambda_2, \phi_1 \cdot \phi_1\right)}}}\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{+45}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{\lambda_1 \cdot R}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq -2.25 \cdot 10^{-200}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.15 \cdot 10^{+45}:\\ \;\;\;\;\lambda_2 \cdot \left(R - \frac{\lambda_1 \cdot R}{\lambda_2}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 -2.25e-200)
   (* phi1 (- R))
   (if (<= phi2 1.15e+45)
     (* lambda2 (- R (/ (* lambda1 R) lambda2)))
     (* R (- phi2 phi1)))))

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 <= -2.25e-200) {
		tmp = phi1 * -R;
	} else if (phi2 <= 1.15e+45) {
		tmp = lambda2 * (R - ((lambda1 * R) / lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= (-2.25d-200)) then
        tmp = phi1 * -r
    else if (phi2 <= 1.15d+45) then
        tmp = lambda2 * (r - ((lambda1 * r) / lambda2))
    else
        tmp = r * (phi2 - phi1)
    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 <= -2.25e-200) {
		tmp = phi1 * -R;
	} else if (phi2 <= 1.15e+45) {
		tmp = lambda2 * (R - ((lambda1 * R) / lambda2));
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 <= -2.25e-200:
		tmp = phi1 * -R
	elif phi2 <= 1.15e+45:
		tmp = lambda2 * (R - ((lambda1 * R) / lambda2))
	else:
		tmp = R * (phi2 - phi1)
	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 <= -2.25e-200)
		tmp = Float64(phi1 * Float64(-R));
	elseif (phi2 <= 1.15e+45)
		tmp = Float64(lambda2 * Float64(R - Float64(Float64(lambda1 * R) / lambda2)));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	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 <= -2.25e-200)
		tmp = phi1 * -R;
	elseif (phi2 <= 1.15e+45)
		tmp = lambda2 * (R - ((lambda1 * R) / lambda2));
	else
		tmp = R * (phi2 - phi1);
	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, -2.25e-200], N[(phi1 * (-R)), $MachinePrecision], If[LessEqual[phi2, 1.15e+45], N[(lambda2 * N[(R - N[(N[(lambda1 * R), $MachinePrecision] / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < -2.2500000000000001e-200

   1. Initial program 58.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 16.9

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

   5. Applied rewrites 16.9%

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

   if -2.2500000000000001e-200 < phi2 < 1.15000000000000006e45

   1. Initial program 62.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 94.0

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 80.0

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

   8. Applied rewrites 80.0%

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

   9. Taylor expanded in lambda2 around inf

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

       1. lower-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. lower--.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. *-commutative N/A

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

       7. lower-*.f64 26.1

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

   11. Applied rewrites 26.1%

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

   if 1.15000000000000006e45 < phi2

   1. Initial program 53.3%

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

   3. Taylor expanded in phi2 around 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 55.3

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

   5. Applied rewrites 55.3%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 55.1

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

   8. Applied rewrites 55.1%

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

Alternative 9: 49.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -8.2 \cdot 10^{+74}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_1 \leq -1.06 \cdot 10^{-56}:\\ \;\;\;\;\phi_2 \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1 \cdot 10^{-252}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 -8.2e+74)
   (* phi1 (- R))
   (if (<= phi1 -1.06e-56)
     (* phi2 R)
     (if (<= phi1 -1e-252) (* lambda2 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 <= -8.2e+74) {
		tmp = phi1 * -R;
	} else if (phi1 <= -1.06e-56) {
		tmp = phi2 * R;
	} else if (phi1 <= -1e-252) {
		tmp = lambda2 * 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 <= (-8.2d+74)) then
        tmp = phi1 * -r
    else if (phi1 <= (-1.06d-56)) then
        tmp = phi2 * r
    else if (phi1 <= (-1d-252)) then
        tmp = lambda2 * 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 <= -8.2e+74) {
		tmp = phi1 * -R;
	} else if (phi1 <= -1.06e-56) {
		tmp = phi2 * R;
	} else if (phi1 <= -1e-252) {
		tmp = lambda2 * 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 <= -8.2e+74:
		tmp = phi1 * -R
	elif phi1 <= -1.06e-56:
		tmp = phi2 * R
	elif phi1 <= -1e-252:
		tmp = lambda2 * 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 <= -8.2e+74)
		tmp = Float64(phi1 * Float64(-R));
	elseif (phi1 <= -1.06e-56)
		tmp = Float64(phi2 * R);
	elseif (phi1 <= -1e-252)
		tmp = Float64(lambda2 * 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 <= -8.2e+74)
		tmp = phi1 * -R;
	elseif (phi1 <= -1.06e-56)
		tmp = phi2 * R;
	elseif (phi1 <= -1e-252)
		tmp = lambda2 * 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, -8.2e+74], N[(phi1 * (-R)), $MachinePrecision], If[LessEqual[phi1, -1.06e-56], N[(phi2 * R), $MachinePrecision], If[LessEqual[phi1, -1e-252], N[(lambda2 * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -8.2000000000000001e74

   1. Initial program 44.8%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if -8.2000000000000001e74 < phi1 < -1.0599999999999999e-56 or -9.99999999999999943e-253 < phi1

   1. Initial program 60.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 11.4

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

   5. Applied rewrites 11.4%

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

   if -1.0599999999999999e-56 < phi1 < -9.99999999999999943e-253

   1. Initial program 69.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 70.4

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

   8. Applied rewrites 70.4%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 29.9

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

   11. Applied rewrites 29.9%

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

Alternative 10: 60.6% accurate, 18.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.4 \cdot 10^{+117}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 (<= lambda1 -2.4e+117) (* lambda1 (- R)) (* R (- phi2 phi1))))

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 (lambda1 <= -2.4e+117) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	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 (lambda1 <= (-2.4d+117)) then
        tmp = lambda1 * -r
    else
        tmp = r * (phi2 - phi1)
    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 (lambda1 <= -2.4e+117) {
		tmp = lambda1 * -R;
	} else {
		tmp = R * (phi2 - phi1);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -2.4e+117:
		tmp = lambda1 * -R
	else:
		tmp = R * (phi2 - phi1)
	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 (lambda1 <= -2.4e+117)
		tmp = Float64(lambda1 * Float64(-R));
	else
		tmp = Float64(R * Float64(phi2 - phi1));
	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 (lambda1 <= -2.4e+117)
		tmp = lambda1 * -R;
	else
		tmp = R * (phi2 - phi1);
	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[lambda1, -2.4e+117], N[(lambda1 * (-R)), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.3999999999999999e117

   1. Initial program 58.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 74.5

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 63.4

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

   8. Applied rewrites 63.4%

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

   9. Taylor expanded in lambda1 around -inf

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 59.6

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

   11. Applied rewrites 59.6%

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

   if -2.3999999999999999e117 < lambda1

   1. Initial program 59.1%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 24.1

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

   5. Applied rewrites 24.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. sub-neg N/A

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

      4. distribute-lft-out-- N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 25.8

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

   8. Applied rewrites 25.8%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -2.4 \cdot 10^{+117}:\\ \;\;\;\;\lambda_1 \cdot \left(-R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 36.9% accurate, 23.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.6 \cdot 10^{+45}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 3.6e+45) (* lambda2 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 <= 3.6e+45) {
		tmp = lambda2 * 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 <= 3.6d+45) then
        tmp = lambda2 * 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 <= 3.6e+45) {
		tmp = lambda2 * 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 <= 3.6e+45:
		tmp = lambda2 * 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 <= 3.6e+45)
		tmp = Float64(lambda2 * 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 <= 3.6e+45)
		tmp = lambda2 * 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, 3.6e+45], N[(lambda2 * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.6e45

   1. Initial program 60.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. unpow2 N/A

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

      5. unpow2 N/A

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

      6. unpow2 N/A

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

      7. unswap-sqr N/A

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

      8. lower-hypot.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lower--.f64 N/A

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

      12. lower-cos.f64 N/A

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

      13. lower-*.f64 76.7

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

   5. Applied rewrites 76.7%

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

   6. Taylor expanded in phi1 around 0

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

      1. lower--.f64 68.6

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in lambda2 around inf

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
   10. Step-by-step derivation

       1. *-commutative N/A

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

       2. lower-*.f64 13.7

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

   11. Applied rewrites 13.7%

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

   if 3.6e45 < phi2

   1. Initial program 53.3%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 45.9

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

   5. Applied rewrites 45.9%

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

Alternative 12: 13.7% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_2 \cdot R \end{array} \]

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 (* lambda2 R))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. unswap-sqr N/A

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

   8. lower-hypot.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 72.0

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

5. Applied rewrites 72.0%

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

6. Taylor expanded in phi1 around 0

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

   1. lower--.f64 65.5

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

8. Applied rewrites 65.5%

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

9. Taylor expanded in lambda2 around inf

   \[\leadsto \color{blue}{R \cdot \lambda_2} \]
10. Step-by-step derivation

    1. *-commutative N/A

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

    2. lower-*.f64 13.0

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

11. Applied rewrites 13.0%

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

Alternative 13: 14.0% accurate, 46.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \lambda_1 \cdot R \end{array} \]

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 (* lambda1 R))

C

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

Python

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

Julia

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

MATLAB

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

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in phi2 around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. unpow2 N/A

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

   5. unpow2 N/A

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

   6. unpow2 N/A

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

   7. unswap-sqr N/A

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

   8. lower-hypot.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. lower-cos.f64 N/A

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

   13. lower-*.f64 72.0

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

5. Applied rewrites 72.0%

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

6. Taylor expanded in lambda1 around inf

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

   1. lower-*.f64 N/A

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

   2. lower-cos.f64 N/A

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

   3. lower-*.f64 12.4

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

8. Applied rewrites 12.4%

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

9. Taylor expanded in phi1 around 0

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

    1. lower-*.f64 10.7

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

11. Applied rewrites 10.7%

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

12. Final simplification 10.7%

    \[\leadsto \lambda_1 \cdot R \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024216 
(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))))))