Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 90.7%

Time: 9.4s

Alternatives: 11

Speedup: 2.3×

• 30.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(-0.5 \cdot \phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(t\_0 \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_2, t\_0, \left(\lambda_2 \cdot \sin \left(\phi_1 \cdot 0.5\right)\right) \cdot \sin \left(\phi_2 \cdot -0.5\right)\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* -0.5 phi1))))
   (if (<= phi2 2.8e-23)
     (* R (hypot (* t_0 (- lambda1 lambda2)) phi1))
     (*
      R
      (hypot
       (fma
        (* (cos (* phi2 0.5)) lambda2)
        t_0
        (* (* lambda2 (sin (* phi1 0.5))) (sin (* phi2 -0.5))))
       (- phi1 phi2))))))

C

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 <= 2.8e-23) {
		tmp = R * hypot((t_0 * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot(fma((cos((phi2 * 0.5)) * lambda2), t_0, ((lambda2 * sin((phi1 * 0.5))) * sin((phi2 * -0.5)))), (phi1 - phi2));
	}
	return tmp;
}

Julia

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 <= 2.8e-23)
		tmp = Float64(R * hypot(Float64(t_0 * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(fma(Float64(cos(Float64(phi2 * 0.5)) * lambda2), t_0, Float64(Float64(lambda2 * sin(Float64(phi1 * 0.5))) * sin(Float64(phi2 * -0.5)))), Float64(phi1 - phi2)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi2, 2.8e-23], N[(R * N[Sqrt[N[(t$95$0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] * t$95$0 + N[(N[(lambda2 * N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sin[N[(phi2 * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.7999999999999997e-23

   1. Initial program 61.9%

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

   3. Taylor expanded in phi2 around 0

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

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 2.7999999999999997e-23 < phi2

   1. Initial program 57.3%

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

   3. Taylor expanded in 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. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 91.7%

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

   9. Applied rewrites 91.7%

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

   11. Applied rewrites 91.7%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 2.8 \cdot 10^{-23}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_2\right), \cos \left(0.5 \cdot \phi_1\right), \sin \left(-0.5 \cdot \phi_2\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 2.8e-23)
   (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
   (*
    R
    (hypot
     (*
      (fma
       (cos (* -0.5 phi2))
       (cos (* 0.5 phi1))
       (* (sin (* -0.5 phi2)) (sin (* 0.5 phi1))))
      lambda2)
     (- phi1 phi2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 2.8e-23) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((fma(cos((-0.5 * phi2)), cos((0.5 * phi1)), (sin((-0.5 * phi2)) * sin((0.5 * phi1)))) * lambda2), (phi1 - phi2));
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 2.8e-23)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(fma(cos(Float64(-0.5 * phi2)), cos(Float64(0.5 * phi1)), Float64(sin(Float64(-0.5 * phi2)) * sin(Float64(0.5 * phi1)))) * lambda2), Float64(phi1 - phi2)));
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 2.8e-23], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $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]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 2.7999999999999997e-23

   1. Initial program 61.9%

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

   3. Taylor expanded in phi2 around 0

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

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if 2.7999999999999997e-23 < phi2

   1. Initial program 57.3%

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

   3. Taylor expanded in 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. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 91.7%

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

Alternative 3: 91.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1000.0)
   (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (* (cos (* -0.5 phi2)) (- lambda1 lambda2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1000.0) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1000.0) {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1000.0:
		tmp = R * math.hypot((math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((math.cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1000.0)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi2)) * Float64(lambda1 - lambda2)), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1000.0)
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((cos((-0.5 * phi2)) * (lambda1 - lambda2)), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1000.0], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1e3

   1. Initial program 58.7%

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

   3. Taylor expanded in phi2 around 0

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

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -1e3 < phi1

   1. Initial program 61.7%

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

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 84.2

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

   5. Applied rewrites 84.2%

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

Alternative 4: 85.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.65 \cdot 10^{+30}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(-0.5 \cdot \phi_2\right) \cdot \lambda_2, \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.65e+30)
   (* R (hypot (* (cos (* -0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (* (cos (* -0.5 phi2)) lambda2) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.65e+30) {
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((cos((-0.5 * phi2)) * lambda2), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.65e+30) {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((Math.cos((-0.5 * phi2)) * lambda2), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.65e+30:
		tmp = R * math.hypot((math.cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((math.cos((-0.5 * phi2)) * lambda2), phi2)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.65e+30)
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(-0.5 * phi2)) * lambda2), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.65e+30)
		tmp = R * hypot((cos((-0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((cos((-0.5 * phi2)) * lambda2), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.65e+30], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(N[Cos[N[(-0.5 * phi2), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.65000000000000013e30

   1. Initial program 61.7%

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

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if 1.65000000000000013e30 < phi2

   1. Initial program 56.7%

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

   3. Taylor expanded in lambda1 around 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. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

   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 74.3%

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

Alternative 5: 80.1% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1 \cdot 10^{+77}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_2, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1e+77)
   (* R (hypot (* (cos (* 0.5 phi1)) lambda2) phi1))
   (* R (hypot (* 1.0 (- lambda1 lambda2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1e+77) {
		tmp = R * hypot((cos((0.5 * phi1)) * lambda2), phi1);
	} else {
		tmp = R * hypot((1.0 * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1e+77) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * lambda2), phi1);
	} else {
		tmp = R * Math.hypot((1.0 * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1e+77:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * lambda2), phi1)
	else:
		tmp = R * math.hypot((1.0 * (lambda1 - lambda2)), phi2)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1e+77)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * lambda2), phi1));
	else
		tmp = Float64(R * hypot(Float64(1.0 * Float64(lambda1 - lambda2)), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1e+77)
		tmp = R * hypot((cos((0.5 * phi1)) * lambda2), phi1);
	else
		tmp = R * hypot((1.0 * (lambda1 - lambda2)), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1e+77], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -9.99999999999999983e76

   1. Initial program 53.7%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. 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. cos-neg-rev N/A

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in lambda1 around 0

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

   8. Applied rewrites 84.9%

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

   if -9.99999999999999983e76 < phi1

   1. Initial program 62.9%

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

   3. Taylor expanded in 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. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   7. Applied rewrites 79.3%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   9. 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 82.3

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

   10. Applied rewrites 82.3%

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

   11. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 74.9%

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 80.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -390000000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \lambda_1, \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -390000000000.0)
   (* R (hypot (* (cos (* 0.5 phi1)) lambda1) phi1))
   (* R (hypot (* 1.0 (- lambda1 lambda2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -390000000000.0) {
		tmp = R * hypot((cos((0.5 * phi1)) * lambda1), phi1);
	} else {
		tmp = R * hypot((1.0 * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -390000000000.0) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * lambda1), phi1);
	} else {
		tmp = R * Math.hypot((1.0 * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -390000000000.0:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * lambda1), phi1)
	else:
		tmp = R * math.hypot((1.0 * (lambda1 - lambda2)), phi2)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -390000000000.0)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * lambda1), phi1));
	else
		tmp = Float64(R * hypot(Float64(1.0 * Float64(lambda1 - lambda2)), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -390000000000.0)
		tmp = R * hypot((cos((0.5 * phi1)) * lambda1), phi1);
	else
		tmp = R * hypot((1.0 * (lambda1 - lambda2)), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -390000000000.0], N[(R * N[Sqrt[N[(N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision] * lambda1), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.9e11

   1. Initial program 58.7%

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

   3. Taylor expanded in phi2 around 0

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

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

      8. lower-cos.f64 N/A

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

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

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

      10. metadata-eval N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in lambda2 around 0

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

   8. Applied rewrites 73.7%

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

   if -3.9e11 < phi1

   1. Initial program 61.7%

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

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 78.5

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

   5. Applied rewrites 78.5%

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   9. 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 84.2

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

   10. Applied rewrites 84.2%

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

   11. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -7.4 \cdot 10^{+47}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -7.4e+47)
   (* (- phi1) (- R (* R (/ phi2 phi1))))
   (* R (hypot (* 1.0 (- lambda1 lambda2)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.4e+47) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = R * hypot((1.0 * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -7.4e+47) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = R * Math.hypot((1.0 * (lambda1 - lambda2)), phi2);
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -7.4e+47:
		tmp = -phi1 * (R - (R * (phi2 / phi1)))
	else:
		tmp = R * math.hypot((1.0 * (lambda1 - lambda2)), phi2)
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -7.4e+47)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))));
	else
		tmp = Float64(R * hypot(Float64(1.0 * Float64(lambda1 - lambda2)), phi2));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -7.4e+47)
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	else
		tmp = R * hypot((1.0 * (lambda1 - lambda2)), phi2);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -7.4e+47], N[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(1.0 * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -7.40000000000000081e47

   1. Initial program 55.2%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(\phi_1 \cdot \left(R + -1 \cdot \frac{R \cdot \phi_2}{\phi_1}\right)\right)} \]
   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. mul-1-neg N/A

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

      3. lower-*.f64 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. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 76.8

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

   5. Applied rewrites 76.8%

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

   if -7.40000000000000081e47 < phi1

   1. Initial program 62.7%

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

   3. Taylor expanded in 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. cos-neg-rev N/A

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

      9. lower-cos.f64 N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 N/A

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

      15. lower--.f64 78.1

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

   5. Applied rewrites 78.1%

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

   7. Applied rewrites 79.4%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \color{blue}{\sqrt{{\cos \left(\frac{1}{2} \cdot \phi_2\right)}^{2} \cdot {\left(\lambda_1 - \lambda_2\right)}^{2} + {\phi_2}^{2}}} \]
   9. 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 83.7

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

   10. Applied rewrites 83.7%

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

   11. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.0%

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 57.8% accurate, 8.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.3 \cdot 10^{+152}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R - R \cdot \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 1.3e+152)
   (* (- phi1) (- R (* R (/ phi2 phi1))))
   (* (- R (* R (/ phi1 phi2))) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.3e+152) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.3d+152) then
        tmp = -phi1 * (r - (r * (phi2 / phi1)))
    else
        tmp = (r - (r * (phi1 / phi2))) * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.3e+152) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.3e+152:
		tmp = -phi1 * (R - (R * (phi2 / phi1)))
	else:
		tmp = (R - (R * (phi1 / phi2))) * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.3e+152)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))));
	else
		tmp = Float64(Float64(R - Float64(R * Float64(phi1 / phi2))) * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.3e+152)
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	else
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 1.3e+152], N[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.3e152

   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. mul-1-neg N/A

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

      3. lower-*.f64 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. fp-cancel-sign-sub-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. lower--.f64 N/A

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

      9. associate-/l* N/A

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

      10. lower-*.f64 N/A

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

      11. lower-/.f64 28.1

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

   5. Applied rewrites 28.1%

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

   if 1.3e152 < phi2

   1. Initial program 54.3%

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

   3. Taylor expanded in 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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 86.7

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

   5. Applied rewrites 86.7%

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

Alternative 9: 56.9% accurate, 9.0× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.7 \cdot 10^{+108}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(R - R \cdot \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.7e+108) (* R (- phi1)) (* (- R (* R (/ phi1 phi2))) phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.7e+108) {
		tmp = R * -phi1;
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.7d+108)) then
        tmp = r * -phi1
    else
        tmp = (r - (r * (phi1 / phi2))) * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.7e+108) {
		tmp = R * -phi1;
	} else {
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.7e+108:
		tmp = R * -phi1
	else:
		tmp = (R - (R * (phi1 / phi2))) * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.7e+108)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(Float64(R - Float64(R * Float64(phi1 / phi2))) * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.7e+108)
		tmp = R * -phi1;
	else
		tmp = (R - (R * (phi1 / phi2))) * phi2;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.7e+108], N[(R * (-phi1)), $MachinePrecision], N[(N[(R - N[(R * N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.6999999999999998e108

   1. Initial program 54.2%

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

   3. Taylor expanded in phi1 around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 82.1

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

   5. Applied rewrites 82.1%

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

   if -3.6999999999999998e108 < phi1

   1. Initial program 62.5%

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

   3. Taylor expanded in 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. fp-cancel-sign-sub-inv N/A

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

      4. metadata-eval N/A

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

      5. *-lft-identity N/A

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

      6. lower--.f64 N/A

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

      7. associate-/l* N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 17.7

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

   5. Applied rewrites 17.7%

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

Alternative 10: 52.1% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -3.1 \cdot 10^{+55}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -3.1e+55) (* R (- phi1)) (* R phi2)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.1e+55) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi1 <= (-3.1d+55)) then
        tmp = r * -phi1
    else
        tmp = r * phi2
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -3.1e+55) {
		tmp = R * -phi1;
	} else {
		tmp = R * phi2;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -3.1e+55:
		tmp = R * -phi1
	else:
		tmp = R * phi2
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -3.1e+55)
		tmp = Float64(R * Float64(-phi1));
	else
		tmp = Float64(R * phi2);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -3.1e+55)
		tmp = R * -phi1;
	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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -3.1e+55], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -3.09999999999999994e55

   1. Initial program 53.8%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   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 69.3

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

   5. Applied rewrites 69.3%

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

   if -3.09999999999999994e55 < phi1

   1. Initial program 63.1%

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

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 15.5

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

   5. Applied rewrites 15.5%

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

Alternative 11: 30.4% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Julia

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])){:}
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.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := N[(R * phi2), $MachinePrecision]

Derivation

1. Initial program 60.8%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 15.3

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

5. Applied rewrites 15.3%

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

Reproduce

herbie shell --seed 2024340 
(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))))))