Equirectangular approximation to distance on a great circle

Percentage Accurate: 58.8% → 91.2%

Time: 10.1s

Alternatives: 15

Speedup: 8.2×

• 30.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 58.8% 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: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot t\_0, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_2\right) \cdot t\_0\right) \cdot \lambda_2 - \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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
 (let* ((t_0 (cos (* 0.5 phi1))))
   (if (<= phi2 2.6e-20)
     (* (hypot (* (- lambda1 lambda2) t_0) phi1) R)
     (*
      (hypot
       (-
        (* (* (cos (* 0.5 phi2)) t_0) lambda2)
        (* (* (sin (* 0.5 phi2)) (sin (* 0.5 phi1))) lambda2))
       (- phi1 phi2))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double tmp;
	if (phi2 <= 2.6e-20) {
		tmp = hypot(((lambda1 - lambda2) * t_0), phi1) * R;
	} else {
		tmp = hypot((((cos((0.5 * phi2)) * t_0) * lambda2) - ((sin((0.5 * phi2)) * sin((0.5 * phi1))) * lambda2)), (phi1 - phi2)) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * phi1));
	double tmp;
	if (phi2 <= 2.6e-20) {
		tmp = Math.hypot(((lambda1 - lambda2) * t_0), phi1) * R;
	} else {
		tmp = Math.hypot((((Math.cos((0.5 * phi2)) * t_0) * lambda2) - ((Math.sin((0.5 * phi2)) * Math.sin((0.5 * phi1))) * lambda2)), (phi1 - phi2)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	tmp = 0
	if phi2 <= 2.6e-20:
		tmp = math.hypot(((lambda1 - lambda2) * t_0), phi1) * R
	else:
		tmp = math.hypot((((math.cos((0.5 * phi2)) * t_0) * lambda2) - ((math.sin((0.5 * phi2)) * math.sin((0.5 * phi1))) * lambda2)), (phi1 - phi2)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	tmp = 0.0
	if (phi2 <= 2.6e-20)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * t_0), phi1) * R);
	else
		tmp = Float64(hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi2)) * t_0) * lambda2) - Float64(Float64(sin(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1))) * lambda2)), Float64(phi1 - phi2)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1));
	tmp = 0.0;
	if (phi2 <= 2.6e-20)
		tmp = hypot(((lambda1 - lambda2) * t_0), phi1) * R;
	else
		tmp = hypot((((cos((0.5 * phi2)) * t_0) * lambda2) - ((sin((0.5 * phi2)) * sin((0.5 * phi1))) * lambda2)), (phi1 - 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.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2.6e-20], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] * lambda2), $MachinePrecision] - N[(N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.59999999999999995e-20

   1. Initial program 61.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if 2.59999999999999995e-20 < phi2

   1. Initial program 61.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 84.4%

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

   9. Applied rewrites 84.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2 - \left(\sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot t\_0, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_2\right) \cdot t\_0 - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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
 (let* ((t_0 (cos (* 0.5 phi1))))
   (if (<= phi2 2.6e-20)
     (* (hypot (* (- lambda1 lambda2) t_0) phi1) R)
     (*
      (hypot
       (*
        (-
         (* (cos (* 0.5 phi2)) t_0)
         (* (sin (* 0.5 phi2)) (sin (* 0.5 phi1))))
        lambda2)
       (- phi1 phi2))
      R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((0.5 * phi1));
	double tmp;
	if (phi2 <= 2.6e-20) {
		tmp = hypot(((lambda1 - lambda2) * t_0), phi1) * R;
	} else {
		tmp = hypot((((cos((0.5 * phi2)) * t_0) - (sin((0.5 * phi2)) * sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((0.5 * phi1));
	double tmp;
	if (phi2 <= 2.6e-20) {
		tmp = Math.hypot(((lambda1 - lambda2) * t_0), phi1) * R;
	} else {
		tmp = Math.hypot((((Math.cos((0.5 * phi2)) * t_0) - (Math.sin((0.5 * phi2)) * Math.sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((0.5 * phi1))
	tmp = 0
	if phi2 <= 2.6e-20:
		tmp = math.hypot(((lambda1 - lambda2) * t_0), phi1) * R
	else:
		tmp = math.hypot((((math.cos((0.5 * phi2)) * t_0) - (math.sin((0.5 * phi2)) * math.sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(0.5 * phi1))
	tmp = 0.0
	if (phi2 <= 2.6e-20)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * t_0), phi1) * R);
	else
		tmp = Float64(hypot(Float64(Float64(Float64(cos(Float64(0.5 * phi2)) * t_0) - Float64(sin(Float64(0.5 * phi2)) * sin(Float64(0.5 * phi1)))) * lambda2), Float64(phi1 - phi2)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((0.5 * phi1));
	tmp = 0.0;
	if (phi2 <= 2.6e-20)
		tmp = hypot(((lambda1 - lambda2) * t_0), phi1) * R;
	else
		tmp = hypot((((cos((0.5 * phi2)) * t_0) - (sin((0.5 * phi2)) * sin((0.5 * phi1)))) * lambda2), (phi1 - 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.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2.6e-20], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision] - N[(N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.59999999999999995e-20

   1. Initial program 61.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if 2.59999999999999995e-20 < phi2

   1. Initial program 61.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   7. Applied rewrites 84.4%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 71.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -0.15:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq -1.32 \cdot 10^{-275}:\\ \;\;\;\;\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1} \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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
 (let* ((t_0 (* (hypot (* (cos (* 0.5 phi2)) lambda2) phi2) R)))
   (if (<= phi1 -0.15)
     (* (hypot (* (cos (* 0.5 phi1)) lambda2) phi1) R)
     (if (<= phi1 -2.4e-176)
       t_0
       (if (<= phi1 -1.32e-275)
         (*
          (*
           (/ (* (cos (* (+ phi1 phi2) 0.5)) (- lambda1 lambda2)) lambda1)
           (- lambda1))
          R)
         t_0)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	double tmp;
	if (phi1 <= -0.15) {
		tmp = hypot((cos((0.5 * phi1)) * lambda2), phi1) * R;
	} else if (phi1 <= -2.4e-176) {
		tmp = t_0;
	} else if (phi1 <= -1.32e-275) {
		tmp = (((cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.hypot((Math.cos((0.5 * phi2)) * lambda2), phi2) * R;
	double tmp;
	if (phi1 <= -0.15) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * lambda2), phi1) * R;
	} else if (phi1 <= -2.4e-176) {
		tmp = t_0;
	} else if (phi1 <= -1.32e-275) {
		tmp = (((Math.cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.hypot((math.cos((0.5 * phi2)) * lambda2), phi2) * R
	tmp = 0
	if phi1 <= -0.15:
		tmp = math.hypot((math.cos((0.5 * phi1)) * lambda2), phi1) * R
	elif phi1 <= -2.4e-176:
		tmp = t_0
	elif phi1 <= -1.32e-275:
		tmp = (((math.cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), phi2) * R)
	tmp = 0.0
	if (phi1 <= -0.15)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda2), phi1) * R);
	elseif (phi1 <= -2.4e-176)
		tmp = t_0;
	elseif (phi1 <= -1.32e-275)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(Float64(phi1 + phi2) * 0.5)) * Float64(lambda1 - lambda2)) / lambda1) * Float64(-lambda1)) * R);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = hypot((cos((0.5 * phi2)) * lambda2), phi2) * R;
	tmp = 0.0;
	if (phi1 <= -0.15)
		tmp = hypot((cos((0.5 * phi1)) * lambda2), phi1) * R;
	elseif (phi1 <= -2.4e-176)
		tmp = t_0;
	elseif (phi1 <= -1.32e-275)
		tmp = (((cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -0.15], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -2.4e-176], t$95$0, If[LessEqual[phi1, -1.32e-275], N[(N[(N[(N[(N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] * (-lambda1)), $MachinePrecision] * R), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -0.149999999999999994

   1. Initial program 54.9%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 64.5%

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

   if -0.149999999999999994 < phi1 < -2.40000000000000006e-176 or -1.31999999999999996e-275 < phi1

   1. Initial program 63.5%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 55.4%

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

   if -2.40000000000000006e-176 < phi1 < -1.31999999999999996e-275

   1. Initial program 63.9%

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

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) + -1 \cdot \frac{\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) - \lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\color{blue}{\lambda_1}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 32.4%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -0.15:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -2.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.32 \cdot 10^{-275}:\\ \;\;\;\;\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1} \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 74.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-176}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq -1.32 \cdot 10^{-275}:\\ \;\;\;\;\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1} \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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
 (let* ((t_0 (* (hypot (* (cos (* 0.5 phi2)) lambda2) (- phi1 phi2)) R)))
   (if (<= phi1 -2.4e-176)
     t_0
     (if (<= phi1 -1.32e-275)
       (*
        (*
         (/ (* (cos (* (+ phi1 phi2) 0.5)) (- lambda1 lambda2)) lambda1)
         (- lambda1))
        R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = hypot((cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R;
	double tmp;
	if (phi1 <= -2.4e-176) {
		tmp = t_0;
	} else if (phi1 <= -1.32e-275) {
		tmp = (((cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.hypot((Math.cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R;
	double tmp;
	if (phi1 <= -2.4e-176) {
		tmp = t_0;
	} else if (phi1 <= -1.32e-275) {
		tmp = (((Math.cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.hypot((math.cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R
	tmp = 0
	if phi1 <= -2.4e-176:
		tmp = t_0
	elif phi1 <= -1.32e-275:
		tmp = (((math.cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), Float64(phi1 - phi2)) * R)
	tmp = 0.0
	if (phi1 <= -2.4e-176)
		tmp = t_0;
	elseif (phi1 <= -1.32e-275)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(Float64(phi1 + phi2) * 0.5)) * Float64(lambda1 - lambda2)) / lambda1) * Float64(-lambda1)) * R);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = hypot((cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R;
	tmp = 0.0;
	if (phi1 <= -2.4e-176)
		tmp = t_0;
	elseif (phi1 <= -1.32e-275)
		tmp = (((cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -2.4e-176], t$95$0, If[LessEqual[phi1, -1.32e-275], N[(N[(N[(N[(N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] * (-lambda1)), $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.40000000000000006e-176 or -1.31999999999999996e-275 < phi1

   1. Initial program 61.2%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 78.3

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

   5. Applied rewrites 78.3%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 76.6%

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

   if -2.40000000000000006e-176 < phi1 < -1.31999999999999996e-275

   1. Initial program 63.9%

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

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) + -1 \cdot \frac{\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 32.4%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) - \lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\color{blue}{\lambda_1}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 32.4%

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

4. Final simplification 71.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.4 \cdot 10^{-176}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.32 \cdot 10^{-275}:\\ \;\;\;\;\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1} \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.02:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 0.02)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) (- lambda1 lambda2)) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 0.02) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 0.02) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 0.02:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 0.02)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 0.02)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * (lambda1 - lambda2)), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, 0.02], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 0.0200000000000000004

   1. Initial program 61.6%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if 0.0200000000000000004 < phi2

   1. Initial program 61.2%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 0.02:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 2.8e-20)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (cos (* 0.5 phi2)) lambda2) (- phi1 phi2)) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 2.8e-20) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot((cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 2.8e-20) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot((Math.cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 2.8e-20:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot((math.cos((0.5 * phi2)) * lambda2), (phi1 - phi2)) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.8e-20)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi2)) * lambda2), Float64(phi1 - phi2)) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 2.8e-20)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot((cos((0.5 * phi2)) * lambda2), (phi1 - 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.
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, 2.8e-20], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.8000000000000003e-20

   1. Initial program 61.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

   if 2.8000000000000003e-20 < phi2

   1. Initial program 61.6%

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

   3. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 80.5

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

   5. Applied rewrites 80.5%

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

   6. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\cos \left(\frac{1}{2} \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 80.6%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 1.3e+15)
   (* (hypot (* (cos (* 0.5 phi1)) lambda2) phi1) R)
   (* (- phi2 phi1) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 1.3e+15) {
		tmp = hypot((cos((0.5 * phi1)) * lambda2), phi1) * R;
	} else {
		tmp = (phi2 - phi1) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 1.3e+15) {
		tmp = Math.hypot((Math.cos((0.5 * phi1)) * lambda2), phi1) * R;
	} else {
		tmp = (phi2 - phi1) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.3e+15:
		tmp = math.hypot((math.cos((0.5 * phi1)) * lambda2), phi1) * R
	else:
		tmp = (phi2 - phi1) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.3e+15)
		tmp = Float64(hypot(Float64(cos(Float64(0.5 * phi1)) * lambda2), phi1) * R);
	else
		tmp = Float64(Float64(phi2 - phi1) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 1.3e+15)
		tmp = hypot((cos((0.5 * phi1)) * lambda2), phi1) * R;
	else
		tmp = (phi2 - 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.
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, 1.3e+15], N[(N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.3e15

   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 lambda1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\lambda_2}^{2} \cdot {\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}^{2} + {\left(\phi_1 - \phi_2\right)}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 76.9

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

   5. Applied rewrites 76.9%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 56.9%

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

   if 1.3e15 < phi2

   1. Initial program 60.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 59.7

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

   5. Applied rewrites 59.7%

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

   7. Applied rewrites 64.4%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 64.5%

       \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 62.9%

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

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq 1.36 \cdot 10^{-272}:\\ \;\;\;\;\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1} \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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
 (let* ((t_0 (* (- phi2 phi1) R)))
   (if (<= phi1 -1.15e-9)
     t_0
     (if (<= phi1 1.36e-272)
       (*
        (*
         (/ (* (cos (* (+ phi1 phi2) 0.5)) (- lambda1 lambda2)) lambda1)
         (- lambda1))
        R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi2 - phi1) * R;
	double tmp;
	if (phi1 <= -1.15e-9) {
		tmp = t_0;
	} else if (phi1 <= 1.36e-272) {
		tmp = (((cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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) :: t_0
    real(8) :: tmp
    t_0 = (phi2 - phi1) * r
    if (phi1 <= (-1.15d-9)) then
        tmp = t_0
    else if (phi1 <= 1.36d-272) then
        tmp = (((cos(((phi1 + phi2) * 0.5d0)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * r
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi2 - phi1) * R;
	double tmp;
	if (phi1 <= -1.15e-9) {
		tmp = t_0;
	} else if (phi1 <= 1.36e-272) {
		tmp = (((Math.cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = (phi2 - phi1) * R
	tmp = 0
	if phi1 <= -1.15e-9:
		tmp = t_0
	elif phi1 <= 1.36e-272:
		tmp = (((math.cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R
	else:
		tmp = t_0
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi2 - phi1) * R)
	tmp = 0.0
	if (phi1 <= -1.15e-9)
		tmp = t_0;
	elseif (phi1 <= 1.36e-272)
		tmp = Float64(Float64(Float64(Float64(cos(Float64(Float64(phi1 + phi2) * 0.5)) * Float64(lambda1 - lambda2)) / lambda1) * Float64(-lambda1)) * R);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = (phi2 - phi1) * R;
	tmp = 0.0;
	if (phi1 <= -1.15e-9)
		tmp = t_0;
	elseif (phi1 <= 1.36e-272)
		tmp = (((cos(((phi1 + phi2) * 0.5)) * (lambda1 - lambda2)) / lambda1) * -lambda1) * R;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -1.15e-9], t$95$0, If[LessEqual[phi1, 1.36e-272], N[(N[(N[(N[(N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] / lambda1), $MachinePrecision] * (-lambda1)), $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.15e-9 or 1.35999999999999996e-272 < phi1

   1. Initial program 59.1%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 29.7

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

   5. Applied rewrites 29.7%

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

   7. Applied rewrites 29.6%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 30.2%

       \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 30.7%

       \[\leadsto \left(\phi_2 - \phi_1\right) \cdot R \]

   if -1.15e-9 < phi1 < 1.35999999999999996e-272

   1. Initial program 67.7%

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

   3. Taylor expanded in lambda1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\lambda_1 \cdot \left(\cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) + -1 \cdot \frac{\lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\lambda_1}\right)\right)\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 30.8%

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

   6. Taylor expanded in lambda1 around 0

      \[\leadsto R \cdot \left(\left(-\lambda_1\right) \cdot \frac{\lambda_1 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right) - \lambda_2 \cdot \cos \left(\frac{1}{2} \cdot \left(\phi_1 + \phi_2\right)\right)}{\color{blue}{\lambda_1}}\right) \]
   7. Step-by-step derivation

   8. Applied rewrites 30.8%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.15 \cdot 10^{-9}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq 1.36 \cdot 10^{-272}:\\ \;\;\;\;\left(\frac{\cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right) \cdot \left(\lambda_1 - \lambda_2\right)}{\lambda_1} \cdot \left(-\lambda_1\right)\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 57.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{+132}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.4 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq -9.5 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(-\lambda_1\right) \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{\phi_1}{R \cdot \phi_2}} - R\right) \cdot \phi_1\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.05e+132)
   (* (- phi2 phi1) R)
   (if (<= phi1 -1.4e-173)
     (* (fma R (/ (- phi1) phi2) R) phi2)
     (if (<= phi1 -9.5e-276)
       (* (* (- lambda1) R) (cos (* 0.5 phi2)))
       (* (- (/ 1.0 (/ phi1 (* R phi2))) R) phi1)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.05e+132) {
		tmp = (phi2 - phi1) * R;
	} else if (phi1 <= -1.4e-173) {
		tmp = fma(R, (-phi1 / phi2), R) * phi2;
	} else if (phi1 <= -9.5e-276) {
		tmp = (-lambda1 * R) * cos((0.5 * phi2));
	} else {
		tmp = ((1.0 / (phi1 / (R * phi2))) - R) * phi1;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.05e+132)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (phi1 <= -1.4e-173)
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * phi2);
	elseif (phi1 <= -9.5e-276)
		tmp = Float64(Float64(Float64(-lambda1) * R) * cos(Float64(0.5 * phi2)));
	else
		tmp = Float64(Float64(Float64(1.0 / Float64(phi1 / Float64(R * phi2))) - R) * phi1);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.05e+132], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -1.4e-173], N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision], If[LessEqual[phi1, -9.5e-276], N[(N[((-lambda1) * R), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / N[(phi1 / N[(R * phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - R), $MachinePrecision] * phi1), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi1 < -1.04999999999999997e132

   1. Initial program 45.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 68.9

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

   5. Applied rewrites 68.9%

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

   7. Applied rewrites 63.5%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 63.5%

       \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 68.9%

       \[\leadsto \left(\phi_2 - \phi_1\right) \cdot R \]

   if -1.04999999999999997e132 < phi1 < -1.39999999999999995e-173

   1. Initial program 63.2%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 37.5

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

   5. Applied rewrites 37.5%

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

   if -1.39999999999999995e-173 < phi1 < -9.49999999999999929e-276

   1. Initial program 62.0%

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

   3. Taylor expanded in lambda1 around -inf

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

      1. mul-1-neg N/A

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

      2. associate-*r* N/A

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

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

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

      4. lower-*.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-+.f64 28.7

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

   5. Applied rewrites 28.7%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 28.7%

      \[\leadsto \left(-R \cdot \lambda_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) \]

   if -9.49999999999999929e-276 < phi1

   1. Initial program 65.0%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 19.7

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

   5. Applied rewrites 19.7%

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

   7. Applied rewrites 21.2%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.05 \cdot 10^{+132}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.4 \cdot 10^{-173}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \mathbf{elif}\;\phi_1 \leq -9.5 \cdot 10^{-276}:\\ \;\;\;\;\left(\left(-\lambda_1\right) \cdot R\right) \cdot \cos \left(0.5 \cdot \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{\frac{\phi_1}{R \cdot \phi_2}} - R\right) \cdot \phi_1\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 59.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;R \leq 2.5 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, \left(-R\right) \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{R}{\phi_2} - \frac{R}{\phi_1}\right) \cdot \phi_2\right) \cdot \left(-\phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (<= R 2.5e+131)
   (fma phi2 R (* (- R) phi1))
   (* (* (- (/ R phi2) (/ R phi1)) phi2) (- phi1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 2.5e+131) {
		tmp = fma(phi2, R, (-R * phi1));
	} else {
		tmp = (((R / phi2) - (R / phi1)) * phi2) * -phi1;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 2.5e+131)
		tmp = fma(phi2, R, Float64(Float64(-R) * phi1));
	else
		tmp = Float64(Float64(Float64(Float64(R / phi2) - Float64(R / phi1)) * phi2) * Float64(-phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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[R, 2.5e+131], N[(phi2 * R + N[((-R) * phi1), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(R / phi2), $MachinePrecision] - N[(R / phi1), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision] * (-phi1)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 2.49999999999999998e131

   1. Initial program 54.4%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 26.6

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

   5. Applied rewrites 26.6%

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 26.7%

       \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 27.2%

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

   if 2.49999999999999998e131 < R

   1. Initial program 97.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in phi2 around inf

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

   8. Applied rewrites 43.1%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;R \leq 2.5 \cdot 10^{+131}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, \left(-R\right) \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\frac{R}{\phi_2} - \frac{R}{\phi_1}\right) \cdot \phi_2\right) \cdot \left(-\phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3.9 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{R}{\phi_1} \cdot \phi_2 - R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, \left(-R\right) \cdot \phi_1\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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) -3.9e+102)
   (* (- (* (/ R phi1) phi2) R) phi1)
   (fma phi2 R (* (- R) phi1))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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) <= -3.9e+102) {
		tmp = (((R / phi1) * phi2) - R) * phi1;
	} else {
		tmp = fma(phi2, R, (-R * phi1));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
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) <= -3.9e+102)
		tmp = Float64(Float64(Float64(Float64(R / phi1) * phi2) - R) * phi1);
	else
		tmp = fma(phi2, R, Float64(Float64(-R) * phi1));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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], -3.9e+102], N[(N[(N[(N[(R / phi1), $MachinePrecision] * phi2), $MachinePrecision] - R), $MachinePrecision] * phi1), $MachinePrecision], N[(phi2 * R + N[((-R) * phi1), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 lambda1 lambda2) < -3.8999999999999998e102

   1. Initial program 50.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 25.8

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

   5. Applied rewrites 25.8%

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

   7. Applied rewrites 27.2%

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

   if -3.8999999999999998e102 < (-.f64 lambda1 lambda2)

   1. Initial program 65.6%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 30.4

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

   5. Applied rewrites 30.4%

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

   7. Applied rewrites 30.4%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 30.5%

       \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 31.6%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 - \lambda_2 \leq -3.9 \cdot 10^{+102}:\\ \;\;\;\;\left(\frac{R}{\phi_1} \cdot \phi_2 - R\right) \cdot \phi_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, \left(-R\right) \cdot \phi_1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;R \leq 6.8 \cdot 10^{+129}:\\ \;\;\;\;\mathsf{fma}\left(\phi_2, R, \left(-R\right) \cdot \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{\phi_2}{\phi_1}, -R\right) \cdot \phi_1\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 (<= R 6.8e+129)
   (fma phi2 R (* (- R) phi1))
   (* (fma R (/ phi2 phi1) (- R)) phi1)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (R <= 6.8e+129) {
		tmp = fma(phi2, R, (-R * phi1));
	} else {
		tmp = fma(R, (phi2 / phi1), -R) * phi1;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (R <= 6.8e+129)
		tmp = fma(phi2, R, Float64(Float64(-R) * phi1));
	else
		tmp = Float64(fma(R, Float64(phi2 / phi1), Float64(-R)) * phi1);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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[R, 6.8e+129], N[(phi2 * R + N[((-R) * phi1), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[(phi2 / phi1), $MachinePrecision] + (-R)), $MachinePrecision] * phi1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if R < 6.80000000000000036e129

   1. Initial program 54.4%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 26.6

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

   5. Applied rewrites 26.6%

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

   7. Applied rewrites 27.6%

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

   8. Taylor expanded in phi1 around 0

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

   10. Applied rewrites 26.7%

       \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
   11. Step-by-step derivation

   12. Applied rewrites 27.2%

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

   if 6.80000000000000036e129 < R

   1. Initial program 97.7%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in phi1 around inf

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

   8. Applied rewrites 42.0%

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

Alternative 13: 53.1% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 5e-43) (* (- phi1) R) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
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 <= 5e-43) {
		tmp = -phi1 * R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= 5d-43) then
        tmp = -phi1 * r
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
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 <= 5e-43) {
		tmp = -phi1 * R;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e-43:
		tmp = -phi1 * R
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e-43)
		tmp = Float64(Float64(-phi1) * R);
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
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 <= 5e-43)
		tmp = -phi1 * R;
	else
		tmp = R * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, 5e-43], N[((-phi1) * R), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.00000000000000019e-43

   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 phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 16.7

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

   5. Applied rewrites 16.7%

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

   if 5.00000000000000019e-43 < phi2

   1. Initial program 64.3%

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

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 56.4

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

   5. Applied rewrites 56.4%

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

4. Final simplification 27.7%

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

Alternative 14: 58.8% accurate, 31.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \left(\phi_2 - \phi_1\right) \cdot R \end{array} \]

FPCore

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

C

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

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = (phi2 - phi1) * r
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

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

   2. lower-*.f64 N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. mul-1-neg N/A

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

   6. unsub-neg N/A

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

   7. lower--.f64 N/A

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

   8. associate-/l* N/A

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

   9. lower-*.f64 N/A

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

   10. lower-/.f64 29.1

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

5. Applied rewrites 29.1%

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

7. Applied rewrites 29.6%

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

8. Taylor expanded in phi1 around 0

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

10. Applied rewrites 27.7%

    \[\leadsto \mathsf{fma}\left(-\phi_1, \color{blue}{R}, \phi_2 \cdot R\right) \]
11. Step-by-step derivation

12. Applied rewrites 28.1%

    \[\leadsto \left(\phi_2 - \phi_1\right) \cdot R \]
13. Add Preprocessing

Alternative 15: 32.1% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.5%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 17.7

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

5. Applied rewrites 17.7%

   \[\leadsto \color{blue}{R \cdot \phi_2} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(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))))))