Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 91.1%

Time: 14.6s

Alternatives: 12

Speedup: 8.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 7e-43)
   (* (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 <= 7e-43) {
		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 <= 7e-43) {
		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 <= 7e-43:
		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 <= 7e-43)
		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 <= 7e-43)
		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, 7e-43], 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 < 6.99999999999999994e-43

   1. Initial program 68.8%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.0

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

   5. Simplified 77.0%

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

   if 6.99999999999999994e-43 < phi2

   1. Initial program 61.8%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. --lowering--.f64 83.0

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

   5. Simplified 83.0%

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

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 7 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 246000000000:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \end{array} \]

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 246000000000.0)
   (* (hypot phi1 (* (- lambda1 lambda2) (cos (* phi1 0.5)))) R)
   (* R (hypot phi2 (* 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 <= 246000000000.0) {
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * hypot(phi2, (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 <= 246000000000.0) {
		tmp = Math.hypot(phi1, ((lambda1 - lambda2) * Math.cos((phi1 * 0.5)))) * R;
	} else {
		tmp = R * Math.hypot(phi2, (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 <= 246000000000.0:
		tmp = math.hypot(phi1, ((lambda1 - lambda2) * math.cos((phi1 * 0.5)))) * R
	else:
		tmp = R * math.hypot(phi2, (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 <= 246000000000.0)
		tmp = Float64(hypot(phi1, Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5)))) * R);
	else
		tmp = Float64(R * hypot(phi2, Float64(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 <= 246000000000.0)
		tmp = hypot(phi1, ((lambda1 - lambda2) * cos((phi1 * 0.5)))) * R;
	else
		tmp = R * hypot(phi2, (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, 246000000000.0], 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[(lambda2 * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.46e11

   1. Initial program 68.8%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.5

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

   5. Simplified 77.5%

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

   if 2.46e11 < phi2

   1. Initial program 60.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. cos-lowering-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. *-lowering-*.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. --lowering--.f64 84.0

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

   5. Simplified 84.0%

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

   6. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. accelerator-lowering-hypot.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-lowering-*.f64 75.8

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

   8. Simplified 75.8%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 246000000000:\\ \;\;\;\;\mathsf{hypot}\left(\phi_1, \left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_2 \cdot \cos \left(\phi_2 \cdot 0.5\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 3.2 \cdot 10^{+138}:\\ \;\;\;\;R \cdot \sqrt{\left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 9.2e-30)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi2 3.2e+138)
     (*
      R
      (sqrt
       (+
        (*
         (+ 0.5 (* 0.5 (cos (+ phi2 phi1))))
         (* (- lambda1 lambda2) (- lambda1 lambda2)))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* 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 <= 9.2e-30) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 3.2e+138) {
		tmp = R * sqrt((((0.5 + (0.5 * cos((phi2 + phi1)))) * ((lambda1 - lambda2) * (lambda1 - lambda2))) + ((phi1 - phi2) * (phi1 - phi2))));
	} 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 <= 9.2e-30) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 3.2e+138) {
		tmp = R * Math.sqrt((((0.5 + (0.5 * Math.cos((phi2 + phi1)))) * ((lambda1 - lambda2) * (lambda1 - lambda2))) + ((phi1 - phi2) * (phi1 - phi2))));
	} 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 <= 9.2e-30:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	elif phi2 <= 3.2e+138:
		tmp = R * math.sqrt((((0.5 + (0.5 * math.cos((phi2 + phi1)))) * ((lambda1 - lambda2) * (lambda1 - lambda2))) + ((phi1 - phi2) * (phi1 - phi2))))
	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 <= 9.2e-30)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi2 <= 3.2e+138)
		tmp = Float64(R * sqrt(Float64(Float64(Float64(0.5 + Float64(0.5 * cos(Float64(phi2 + phi1)))) * Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2))) + Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	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 <= 9.2e-30)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	elseif (phi2 <= 3.2e+138)
		tmp = R * sqrt((((0.5 + (0.5 * cos((phi2 + phi1)))) * ((lambda1 - lambda2) * (lambda1 - lambda2))) + ((phi1 - phi2) * (phi1 - phi2))));
	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, 9.2e-30], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 3.2e+138], N[(R * N[Sqrt[N[(N[(N[(0.5 + N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 9.19999999999999937e-30

   1. Initial program 68.2%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.4

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

   5. Simplified 77.4%

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

   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. --lowering--.f64 75.2

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

   8. Simplified 75.2%

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

   if 9.19999999999999937e-30 < phi2 < 3.2000000000000001e138

   1. Initial program 69.3%

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

      1. swap-sqr N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

   4. Applied egg-rr 69.3%

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

   if 3.2000000000000001e138 < phi2

   1. Initial program 55.3%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto R \cdot \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      8. /-lowering-/.f64 77.3

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right) \]

   5. Simplified 77.3%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 83.1

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

   8. Simplified 83.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 9.2 \cdot 10^{-30}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 3.2 \cdot 10^{+138}:\\ \;\;\;\;R \cdot \sqrt{\left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 82.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right), \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 1e-29)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (if (<= phi2 1.75e+139)
     (*
      R
      (sqrt
       (fma
        (+ 0.5 (* 0.5 (cos (+ phi2 phi1))))
        (* (- lambda1 lambda2) (- lambda1 lambda2))
        (* (- phi1 phi2) (- phi1 phi2)))))
     (* 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 <= 1e-29) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else if (phi2 <= 1.75e+139) {
		tmp = R * sqrt(fma((0.5 + (0.5 * cos((phi2 + phi1)))), ((lambda1 - lambda2) * (lambda1 - lambda2)), ((phi1 - phi2) * (phi1 - phi2))));
	} 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 <= 1e-29)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	elseif (phi2 <= 1.75e+139)
		tmp = Float64(R * sqrt(fma(Float64(0.5 + Float64(0.5 * cos(Float64(phi2 + phi1)))), Float64(Float64(lambda1 - lambda2) * Float64(lambda1 - lambda2)), Float64(Float64(phi1 - phi2) * Float64(phi1 - phi2)))));
	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, 1e-29], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.75e+139], N[(R * N[Sqrt[N[(N[(0.5 + N[(0.5 * N[Cos[N[(phi2 + phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] + N[(N[(phi1 - phi2), $MachinePrecision] * N[(phi1 - phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 9.99999999999999943e-30

   1. Initial program 68.2%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.4

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

   5. Simplified 77.4%

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

   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. --lowering--.f64 75.2

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

   8. Simplified 75.2%

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

   if 9.99999999999999943e-30 < phi2 < 1.74999999999999989e139

   1. Initial program 69.3%

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

      1. swap-sqr N/A

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

      2. *-commutative N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

   4. Applied egg-rr 69.3%

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

   if 1.74999999999999989e139 < phi2

   1. Initial program 55.3%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto R \cdot \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      8. /-lowering-/.f64 77.3

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right) \]

   5. Simplified 77.3%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 83.1

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

   8. Simplified 83.1%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 10^{-29}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{+139}:\\ \;\;\;\;R \cdot \sqrt{\mathsf{fma}\left(0.5 + 0.5 \cdot \cos \left(\phi_2 + \phi_1\right), \left(\lambda_1 - \lambda_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3850000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{\phi_2 \cdot R}{\phi_1} - R\right)\\ \end{array} \end{array} \]

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 3850000.0)
   (* R (hypot phi1 (- lambda1 lambda2)))
   (* phi1 (- (/ (* phi2 R) phi1) 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 <= 3850000.0) {
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = phi1 * (((phi2 * R) / phi1) - R);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 3850000.0) {
		tmp = R * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = phi1 * (((phi2 * R) / phi1) - 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 <= 3850000.0:
		tmp = R * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = phi1 * (((phi2 * R) / phi1) - 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 <= 3850000.0)
		tmp = Float64(R * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(phi1 * Float64(Float64(Float64(phi2 * R) / phi1) - 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 <= 3850000.0)
		tmp = R * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = phi1 * (((phi2 * R) / phi1) - 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, 3850000.0], N[(R * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 3.85e6

   1. Initial program 68.8%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.5

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

   5. Simplified 77.5%

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

   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. --lowering--.f64 74.6

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

   8. Simplified 74.6%

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

   if 3.85e6 < phi2

   1. Initial program 60.5%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto 0 - \phi_1 \cdot \left(R + \color{blue}{\left(\mathsf{neg}\left(\frac{R \cdot \phi_2}{\phi_1}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto 0 - \phi_1 \cdot \color{blue}{\left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto 0 - \phi_1 \cdot \color{blue}{\left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto 0 - \phi_1 \cdot \left(R - \color{blue}{\frac{R \cdot \phi_2}{\phi_1}}\right) \]

      9. *-commutative N/A

         \[\leadsto 0 - \phi_1 \cdot \left(R - \frac{\color{blue}{\phi_2 \cdot R}}{\phi_1}\right) \]

      10. *-lowering-*.f64 65.6

          \[\leadsto 0 - \phi_1 \cdot \left(R - \frac{\color{blue}{\phi_2 \cdot R}}{\phi_1}\right) \]

   5. Simplified 65.6%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3850000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{\phi_2 \cdot R}{\phi_1} - R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 61.8% accurate, 2.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.6 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 21000:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{\phi_2 \cdot R}{\phi_1} - R\right)\\ \end{array} \end{array} \]

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 5.6e-151)
   (* R (hypot phi1 lambda2))
   (if (<= phi2 21000.0)
     (* R (* lambda2 (- 1.0 (/ lambda1 lambda2))))
     (* phi1 (- (/ (* phi2 R) phi1) 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 <= 5.6e-151) {
		tmp = R * hypot(phi1, lambda2);
	} else if (phi2 <= 21000.0) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = phi1 * (((phi2 * R) / phi1) - R);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.6e-151) {
		tmp = R * Math.hypot(phi1, lambda2);
	} else if (phi2 <= 21000.0) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	} else {
		tmp = phi1 * (((phi2 * R) / phi1) - 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 <= 5.6e-151:
		tmp = R * math.hypot(phi1, lambda2)
	elif phi2 <= 21000.0:
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)))
	else:
		tmp = phi1 * (((phi2 * R) / phi1) - 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 <= 5.6e-151)
		tmp = Float64(R * hypot(phi1, lambda2));
	elseif (phi2 <= 21000.0)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - Float64(lambda1 / lambda2))));
	else
		tmp = Float64(phi1 * Float64(Float64(Float64(phi2 * R) / phi1) - 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 <= 5.6e-151)
		tmp = R * hypot(phi1, lambda2);
	elseif (phi2 <= 21000.0)
		tmp = R * (lambda2 * (1.0 - (lambda1 / lambda2)));
	else
		tmp = phi1 * (((phi2 * R) / phi1) - 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, 5.6e-151], N[(R * N[Sqrt[phi1 ^ 2 + lambda2 ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 21000.0], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(phi1 * N[(N[(N[(phi2 * R), $MachinePrecision] / phi1), $MachinePrecision] - R), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 5.6000000000000002e-151

   1. Initial program 66.5%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 75.0

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

   5. Simplified 75.0%

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

   6. Taylor expanded in lambda1 around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. unpow2 N/A

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

      4. unpow2 N/A

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

      5. unswap-sqr N/A

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

      6. accelerator-lowering-hypot.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. cos-lowering-cos.f64 N/A

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

      9. *-lowering-*.f64 56.0

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

   8. Simplified 56.0%

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

   9. Taylor expanded in phi1 around 0

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

   11. Simplified 56.0%

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

   if 5.6000000000000002e-151 < phi2 < 21000

   1. Initial program 81.2%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 90.8

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

   5. Simplified 90.8%

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

   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. --lowering--.f64 82.3

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

   8. Simplified 82.3%

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

   9. Taylor expanded in lambda2 around inf

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

       1. *-lowering-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. --lowering--.f64 N/A

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

       5. /-lowering-/.f64 46.2

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

   11. Simplified 46.2%

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

   if 21000 < phi2

   1. Initial program 60.5%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto 0 - \color{blue}{\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      5. mul-1-neg N/A

         \[\leadsto 0 - \phi_1 \cdot \left(R + \color{blue}{\left(\mathsf{neg}\left(\frac{R \cdot \phi_2}{\phi_1}\right)\right)}\right) \]

      6. unsub-neg N/A

         \[\leadsto 0 - \phi_1 \cdot \color{blue}{\left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      7. --lowering--.f64 N/A

         \[\leadsto 0 - \phi_1 \cdot \color{blue}{\left(R - \frac{R \cdot \phi_2}{\phi_1}\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto 0 - \phi_1 \cdot \left(R - \color{blue}{\frac{R \cdot \phi_2}{\phi_1}}\right) \]

      9. *-commutative N/A

         \[\leadsto 0 - \phi_1 \cdot \left(R - \frac{\color{blue}{\phi_2 \cdot R}}{\phi_1}\right) \]

      10. *-lowering-*.f64 65.6

          \[\leadsto 0 - \phi_1 \cdot \left(R - \frac{\color{blue}{\phi_2 \cdot R}}{\phi_1}\right) \]

   5. Simplified 65.6%

      \[\leadsto \color{blue}{0 - \phi_1 \cdot \left(R - \frac{\phi_2 \cdot R}{\phi_1}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.6 \cdot 10^{-151}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1, \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 21000:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_1 \cdot \left(\frac{\phi_2 \cdot R}{\phi_1} - R\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 61.8% accurate, 8.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \end{array} \]

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 lambda2) -1e+151)
   (* R (* lambda2 (- 1.0 (/ 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 ((lambda1 - lambda2) <= -1e+151) {
		tmp = R * (lambda2 * (1.0 - (lambda1 / 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 ((lambda1 - lambda2) <= (-1d+151)) then
        tmp = r * (lambda2 * (1.0d0 - (lambda1 / 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 ((lambda1 - lambda2) <= -1e+151) {
		tmp = R * (lambda2 * (1.0 - (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 (lambda1 - lambda2) <= -1e+151:
		tmp = R * (lambda2 * (1.0 - (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 (Float64(lambda1 - lambda2) <= -1e+151)
		tmp = Float64(R * Float64(lambda2 * Float64(1.0 - 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 ((lambda1 - lambda2) <= -1e+151)
		tmp = R * (lambda2 * (1.0 - (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[N[(lambda1 - lambda2), $MachinePrecision], -1e+151], N[(R * N[(lambda2 * N[(1.0 - N[(lambda1 / lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -1.00000000000000002e151

   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 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.6

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

   5. Simplified 77.6%

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

   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. --lowering--.f64 73.5

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

   8. Simplified 73.5%

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

   9. Taylor expanded in lambda2 around inf

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

       1. *-lowering-*.f64 N/A

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

       2. mul-1-neg N/A

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

       3. unsub-neg N/A

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

       4. --lowering--.f64 N/A

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

       5. /-lowering-/.f64 67.9

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

   11. Simplified 67.9%

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

   if -1.00000000000000002e151 < (-.f64 lambda1 lambda2)

   1. Initial program 70.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto R \cdot \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      8. /-lowering-/.f64 33.6

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right) \]

   5. Simplified 33.6%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 35.1

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

   8. Simplified 35.1%

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

4. Final simplification 42.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -1 \cdot 10^{+151}:\\ \;\;\;\;R \cdot \left(\lambda_2 \cdot \left(1 - \frac{\lambda_1}{\lambda_2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.7% accurate, 13.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+133}:\\ \;\;\;\;\lambda_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \end{array} \]

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 (<= lambda1 -1.65e+133)
   (* lambda1 (- 0.0 R))
   (if (<= lambda1 3.3e-21) (* R (- phi2 phi1)) (* lambda2 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 (lambda1 <= -1.65e+133) {
		tmp = lambda1 * (0.0 - R);
	} else if (lambda1 <= 3.3e-21) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = lambda2 * 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 (lambda1 <= (-1.65d+133)) then
        tmp = lambda1 * (0.0d0 - r)
    else if (lambda1 <= 3.3d-21) then
        tmp = r * (phi2 - phi1)
    else
        tmp = lambda2 * 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 (lambda1 <= -1.65e+133) {
		tmp = lambda1 * (0.0 - R);
	} else if (lambda1 <= 3.3e-21) {
		tmp = R * (phi2 - phi1);
	} else {
		tmp = lambda2 * 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 lambda1 <= -1.65e+133:
		tmp = lambda1 * (0.0 - R)
	elif lambda1 <= 3.3e-21:
		tmp = R * (phi2 - phi1)
	else:
		tmp = lambda2 * 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 (lambda1 <= -1.65e+133)
		tmp = Float64(lambda1 * Float64(0.0 - R));
	elseif (lambda1 <= 3.3e-21)
		tmp = Float64(R * Float64(phi2 - phi1));
	else
		tmp = Float64(lambda2 * 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 (lambda1 <= -1.65e+133)
		tmp = lambda1 * (0.0 - R);
	elseif (lambda1 <= 3.3e-21)
		tmp = R * (phi2 - phi1);
	else
		tmp = lambda2 * 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[lambda1, -1.65e+133], N[(lambda1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], If[LessEqual[lambda1, 3.3e-21], N[(R * N[(phi2 - phi1), $MachinePrecision]), $MachinePrecision], N[(lambda2 * R), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if lambda1 < -1.65e133

   1. Initial program 56.0%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 60.9

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

   5. Simplified 60.9%

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

   6. Taylor expanded in phi2 around 0

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

      1. cos-lowering-cos.f64 N/A

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

      2. *-lowering-*.f64 61.2

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

   8. Simplified 61.2%

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

   9. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]

       3. --lowering--.f64 N/A

          \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]

       4. *-lowering-*.f64 70.0

          \[\leadsto 0 - \color{blue}{R \cdot \lambda_1} \]

   11. Simplified 70.0%

       \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]
   12. Step-by-step derivation

       1. sub0-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f64 N/A

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

       5. neg-lowering-neg.f64 70.0

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

   13. Applied egg-rr 70.0%

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

   if -1.65e133 < lambda1 < 3.30000000000000009e-21

   1. Initial program 72.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      3. --lowering--.f64 N/A

         \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto R \cdot \left(0 - \color{blue}{\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 + \color{blue}{\left(\mathsf{neg}\left(\frac{\phi_2}{\phi_1}\right)\right)}\right)\right) \]

      6. unsub-neg N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \color{blue}{\left(1 - \frac{\phi_2}{\phi_1}\right)}\right) \]

      8. /-lowering-/.f64 41.1

         \[\leadsto R \cdot \left(0 - \phi_1 \cdot \left(1 - \color{blue}{\frac{\phi_2}{\phi_1}}\right)\right) \]

   5. Simplified 41.1%

      \[\leadsto R \cdot \color{blue}{\left(0 - \phi_1 \cdot \left(1 - \frac{\phi_2}{\phi_1}\right)\right)} \]

   6. Taylor expanded in phi1 around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

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

      3. remove-double-neg N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. --lowering--.f64 42.5

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

   8. Simplified 42.5%

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

   if 3.30000000000000009e-21 < lambda1

   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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 71.8

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

   5. Simplified 71.8%

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

   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. --lowering--.f64 67.0

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

   8. Simplified 67.0%

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

   9. Taylor expanded in lambda2 around inf

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

   11. Simplified 11.1%

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

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -1.65 \cdot 10^{+133}:\\ \;\;\;\;\lambda_1 \cdot \left(0 - R\right)\\ \mathbf{elif}\;\lambda_1 \leq 3.3 \cdot 10^{-21}:\\ \;\;\;\;R \cdot \left(\phi_2 - \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.2% accurate, 18.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5 \cdot 10^{-29}:\\ \;\;\;\;\lambda_1 \cdot \left(0 - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 2.5e-29) (* lambda1 (- 0.0 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 <= 2.5e-29) {
		tmp = lambda1 * (0.0 - 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 <= 2.5d-29) then
        tmp = lambda1 * (0.0d0 - 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 <= 2.5e-29) {
		tmp = lambda1 * (0.0 - 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 <= 2.5e-29:
		tmp = lambda1 * (0.0 - 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 <= 2.5e-29)
		tmp = Float64(lambda1 * Float64(0.0 - 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 <= 2.5e-29)
		tmp = lambda1 * (0.0 - 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, 2.5e-29], N[(lambda1 * N[(0.0 - R), $MachinePrecision]), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.49999999999999993e-29

   1. Initial program 68.2%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. *-lowering-*.f64 N/A

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

      6. cos-lowering-cos.f64 N/A

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

      7. *-lowering-*.f64 N/A

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

      8. +-lowering-+.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 24.1

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

   5. Simplified 24.1%

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

   6. Taylor expanded in phi2 around 0

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

      1. cos-lowering-cos.f64 N/A

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

      2. *-lowering-*.f64 22.1

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

   8. Simplified 22.1%

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

   9. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. neg-sub0 N/A

          \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]

       3. --lowering--.f64 N/A

          \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]

       4. *-lowering-*.f64 21.9

          \[\leadsto 0 - \color{blue}{R \cdot \lambda_1} \]

   11. Simplified 21.9%

       \[\leadsto \color{blue}{0 - R \cdot \lambda_1} \]
   12. Step-by-step derivation

       1. sub0-neg N/A

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

       2. *-commutative N/A

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

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

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

       4. *-lowering-*.f64 N/A

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

       5. neg-lowering-neg.f64 21.9

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

   13. Applied egg-rr 21.9%

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

   if 2.49999999999999993e-29 < phi2

   1. Initial program 62.9%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   2. 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. *-lowering-*.f64 54.0

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

   5. Simplified 54.0%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.5 \cdot 10^{-29}:\\ \;\;\;\;\lambda_1 \cdot \left(0 - R\right)\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-40}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

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 1.9e-40) (* 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 <= 1.9e-40) {
		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 <= 1.9d-40) 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 <= 1.9e-40) {
		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 <= 1.9e-40:
		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 <= 1.9e-40)
		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 <= 1.9e-40)
		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, 1.9e-40], N[(lambda2 * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.8999999999999999e-40

   1. Initial program 68.9%

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

   3. Taylor expanded in phi2 around 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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 77.2

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

   5. Simplified 77.2%

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

   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. --lowering--.f64 75.1

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

   8. Simplified 75.1%

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

   9. Taylor expanded in lambda2 around inf

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

   11. Simplified 14.8%

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

   if 1.8999999999999999e-40 < phi2

   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 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. *-lowering-*.f64 52.6

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

   5. Simplified 52.6%

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

Alternative 11: 14.4% 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 66.7%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 70.0

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

5. Simplified 70.0%

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

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. --lowering--.f64 67.8

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

8. Simplified 67.8%

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

9. Taylor expanded in lambda2 around inf

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

11. Simplified 15.1%

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

Alternative 12: 14.3% 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 66.7%

   \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
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. *-lowering-*.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. accelerator-lowering-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. *-lowering-*.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. --lowering--.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. cos-lowering-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. *-lowering-*.f64 70.0

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

5. Simplified 70.0%

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

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. --lowering--.f64 67.8

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

8. Simplified 67.8%

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

9. Taylor expanded in lambda1 around inf

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

11. Simplified 15.0%

    \[\leadsto \color{blue}{\lambda_1} \cdot R \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024196 
(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))))))