Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.9% → 91.7%

Time: 9.6s

Alternatives: 13

Speedup: 2.5×

• 30.2% 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: 59.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 3.5 \cdot 10^{+35}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \mathsf{fma}\left(\cos \left(-0.5 \cdot \phi_1\right), \lambda_1 - \lambda_2, \left(0.5 \cdot \phi_2\right) \cdot \left(\sin \left(-0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

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 (<= phi2 3.5e+35)
   (*
    R
    (hypot
     (- phi1 phi2)
     (fma
      (cos (* -0.5 phi1))
      (- lambda1 lambda2)
      (* (* 0.5 phi2) (* (sin (* -0.5 phi1)) (- lambda1 lambda2))))))
   (* 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 (phi2 <= 3.5e+35) {
		tmp = R * hypot((phi1 - phi2), fma(cos((-0.5 * phi1)), (lambda1 - lambda2), ((0.5 * phi2) * (sin((-0.5 * phi1)) * (lambda1 - lambda2)))));
	} else {
		tmp = R * hypot((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 (phi2 <= 3.5e+35)
		tmp = Float64(R * hypot(Float64(phi1 - phi2), fma(cos(Float64(-0.5 * phi1)), Float64(lambda1 - lambda2), Float64(Float64(0.5 * phi2) * Float64(sin(Float64(-0.5 * phi1)) * Float64(lambda1 - lambda2))))));
	else
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi2)) * Float64(lambda1 - lambda2)), 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, 3.5e+35], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(N[Cos[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision] + N[(N[(0.5 * phi2), $MachinePrecision] * N[(N[Sin[N[(-0.5 * phi1), $MachinePrecision]], $MachinePrecision] * N[(lambda1 - lambda2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 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 phi2 < 3.5000000000000001e35

   1. Initial program 52.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 50.0%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 89.7

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

   7. Applied rewrites 89.7%

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

   if 3.5000000000000001e35 < phi2

   1. Initial program 47.7%

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

   3. Taylor expanded in phi1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 82.4

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

   5. Applied rewrites 82.4%

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

Alternative 2: 90.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1700000000:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_2\right)\\ \end{array} \end{array} \]

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 -1700000000.0)
   (* 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 <= -1700000000.0) {
		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 <= -1700000000.0) {
		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 <= -1700000000.0:
		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 <= -1700000000.0)
		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 <= -1700000000.0)
		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, -1700000000.0], 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 < -1.7e9

   1. Initial program 45.8%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   if -1.7e9 < phi1

   1. Initial program 52.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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. 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 77.5

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

   5. Applied rewrites 77.5%

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

Alternative 3: 90.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5 \cdot 10^{-112}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi2 5e-112)
   (* R (hypot (* (cos (* 0.5 phi1)) (- lambda1 lambda2)) phi1))
   (* R (hypot (- phi1 phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-112) {
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5e-112) {
		tmp = R * Math.hypot((Math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	} else {
		tmp = R * Math.hypot((phi1 - phi2), (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5e-112:
		tmp = R * math.hypot((math.cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1)
	else:
		tmp = R * math.hypot((phi1 - phi2), (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5e-112)
		tmp = Float64(R * hypot(Float64(cos(Float64(0.5 * phi1)) * Float64(lambda1 - lambda2)), phi1));
	else
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5e-112)
		tmp = R * hypot((cos((0.5 * phi1)) * (lambda1 - lambda2)), phi1);
	else
		tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi2, 5e-112], 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[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.00000000000000044e-112

   1. Initial program 52.9%

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

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

   if 5.00000000000000044e-112 < phi2

   1. Initial program 46.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 43.1%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 71.5

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 81.7%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 86.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -9 \cdot 10^{+250}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda1 -9e+250)
   (* R (hypot (* lambda1 (cos (* 0.5 phi1))) phi1))
   (* R (hypot (- phi1 phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -9e+250) {
		tmp = R * hypot((lambda1 * cos((0.5 * phi1))), phi1);
	} else {
		tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -9e+250) {
		tmp = R * Math.hypot((lambda1 * Math.cos((0.5 * phi1))), phi1);
	} else {
		tmp = R * Math.hypot((phi1 - phi2), (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -9e+250:
		tmp = R * math.hypot((lambda1 * math.cos((0.5 * phi1))), phi1)
	else:
		tmp = R * math.hypot((phi1 - phi2), (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -9e+250)
		tmp = Float64(R * hypot(Float64(lambda1 * cos(Float64(0.5 * phi1))), phi1));
	else
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -9e+250)
		tmp = R * hypot((lambda1 * cos((0.5 * phi1))), phi1);
	else
		tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda1, -9e+250], N[(R * N[Sqrt[N[(lambda1 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -8.99999999999999993e250

   1. Initial program 49.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 93.8

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

   5. Applied rewrites 93.8%

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

   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 93.8%

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

   if -8.99999999999999993e250 < lambda1

   1. Initial program 51.0%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 48.9%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 83.3

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

   7. Applied rewrites 83.3%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 84.2%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 82.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -6.2 \cdot 10^{-17}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, -\lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -6.2e-17)
   (* R (hypot (- phi1 phi2) (- lambda2)))
   (* R (hypot (- phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.2e-17) {
		tmp = R * hypot((phi1 - phi2), -lambda2);
	} else {
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -6.2e-17) {
		tmp = R * Math.hypot((phi1 - phi2), -lambda2);
	} else {
		tmp = R * Math.hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -6.2e-17:
		tmp = R * math.hypot((phi1 - phi2), -lambda2)
	else:
		tmp = R * math.hypot(-phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -6.2e-17)
		tmp = Float64(R * hypot(Float64(phi1 - phi2), Float64(-lambda2)));
	else
		tmp = Float64(R * hypot(Float64(-phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -6.2e-17)
		tmp = R * hypot((phi1 - phi2), -lambda2);
	else
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -6.2e-17], N[(R * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + (-lambda2) ^ 2], $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[(-phi2) ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -6.1999999999999997e-17

   1. Initial program 48.1%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 54.8%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 87.1

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

   7. Applied rewrites 87.1%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 83.6%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]

   11. Taylor expanded in lambda1 around 0

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

   13. Applied rewrites 80.6%

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

   if -6.1999999999999997e-17 < phi1

   1. Initial program 51.9%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 46.3%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 82.5

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

   7. Applied rewrites 82.5%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 83.4%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]

   11. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 69.6

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

   13. Applied rewrites 69.6%

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

Alternative 6: 80.6% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(-1 + \frac{\phi_2}{\phi_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.1e+51)
   (* R (* phi1 (+ -1.0 (/ phi2 phi1))))
   (* R (hypot (- phi2) (- lambda1 lambda2)))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.1e+51) {
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	} else {
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.1e+51) {
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	} else {
		tmp = R * Math.hypot(-phi2, (lambda1 - lambda2));
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.1e+51:
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)))
	else:
		tmp = R * math.hypot(-phi2, (lambda1 - lambda2))
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.1e+51)
		tmp = Float64(R * Float64(phi1 * Float64(-1.0 + Float64(phi2 / phi1))));
	else
		tmp = Float64(R * hypot(Float64(-phi2), Float64(lambda1 - lambda2)));
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.1e+51)
		tmp = R * (phi1 * (-1.0 + (phi2 / phi1)));
	else
		tmp = R * hypot(-phi2, (lambda1 - lambda2));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.1e+51], N[(R * N[(phi1 * N[(-1.0 + N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[Sqrt[(-phi2) ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.09999999999999996e51

   1. Initial program 41.9%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 74.8

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

   5. Applied rewrites 74.8%

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

   if -1.09999999999999996e51 < phi1

   1. Initial program 53.6%

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

      1. lift-sqrt.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. pow2 N/A

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

      7. pow-to-exp N/A

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

      8. exp-lft-sqr N/A

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

      9. lower-hypot.f64 N/A

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

   4. Applied rewrites 46.1%

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

   5. Taylor expanded in phi2 around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-cos.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. associate-*r* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower--.f64 82.8

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

   7. Applied rewrites 82.8%

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

   8. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 82.5%

       \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]

   11. Taylor expanded in phi1 around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 68.9

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

   13. Applied rewrites 68.9%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{+51}:\\ \;\;\;\;R \cdot \left(\phi_1 \cdot \left(-1 + \frac{\phi_2}{\phi_1}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(-\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \lambda_2\right) \end{array} \]

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 (hypot (- phi1 phi2) (- lambda1 lambda2))))

C

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

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 * Math.hypot((phi1 - phi2), (lambda1 - lambda2));
}

Python

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

Julia

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

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R, lambda1, lambda2, phi1, phi2)
	tmp = R * hypot((phi1 - phi2), (lambda1 - lambda2));
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 * N[Sqrt[N[(phi1 - phi2), $MachinePrecision] ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 50.9%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. pow2 N/A

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

   7. pow-to-exp N/A

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

   8. exp-lft-sqr N/A

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

   9. lower-hypot.f64 N/A

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

4. Applied rewrites 48.6%

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

5. Taylor expanded in phi2 around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-cos.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower--.f64 N/A

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

   6. associate-*r* N/A

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

   7. lower-*.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. lower--.f64 83.7

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

7. Applied rewrites 83.7%

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

8. Taylor expanded in phi1 around 0

   \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
9. Step-by-step derivation

10. Applied rewrites 83.5%

    \[\leadsto R \cdot \mathsf{hypot}\left(\phi_1 - \phi_2, \lambda_1 - \color{blue}{\lambda_2}\right) \]
11. Add Preprocessing

Alternative 8: 58.5% accurate, 5.6× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.85 \cdot 10^{-250}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.75 \cdot 10^{-153}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(0.0026041666666666665, \lambda_2 \cdot \left(\phi_1 \cdot \phi_1\right), -0.125 \cdot \lambda_2\right), \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 10^{+16}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - \frac{R \cdot \phi_2}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(1 - \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 1.85e-250)
   (* (- phi1) (- R (* R (/ phi2 phi1))))
   (if (<= phi2 1.75e-153)
     (*
      R
      (fma
       (* phi1 phi1)
       (fma 0.0026041666666666665 (* lambda2 (* phi1 phi1)) (* -0.125 lambda2))
       lambda2))
     (if (<= phi2 1e+16)
       (* (- phi1) (- R (/ (* R phi2) phi1)))
       (* R (* (- 1.0 (/ 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 <= 1.85e-250) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else if (phi2 <= 1.75e-153) {
		tmp = R * fma((phi1 * phi1), fma(0.0026041666666666665, (lambda2 * (phi1 * phi1)), (-0.125 * lambda2)), lambda2);
	} else if (phi2 <= 1e+16) {
		tmp = -phi1 * (R - ((R * phi2) / phi1));
	} else {
		tmp = R * ((1.0 - (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 <= 1.85e-250)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))));
	elseif (phi2 <= 1.75e-153)
		tmp = Float64(R * fma(Float64(phi1 * phi1), fma(0.0026041666666666665, Float64(lambda2 * Float64(phi1 * phi1)), Float64(-0.125 * lambda2)), lambda2));
	elseif (phi2 <= 1e+16)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(Float64(R * phi2) / phi1)));
	else
		tmp = Float64(R * Float64(Float64(1.0 - Float64(phi1 / phi2)) * 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, 1.85e-250], N[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.75e-153], N[(R * N[(N[(phi1 * phi1), $MachinePrecision] * N[(0.0026041666666666665 * N[(lambda2 * N[(phi1 * phi1), $MachinePrecision]), $MachinePrecision] + N[(-0.125 * lambda2), $MachinePrecision]), $MachinePrecision] + lambda2), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1e+16], N[((-phi1) * N[(R - N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(R * N[(N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < 1.8499999999999999e-250

   1. Initial program 54.0%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 21.2

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

   5. Applied rewrites 21.2%

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

   if 1.8499999999999999e-250 < phi2 < 1.7499999999999999e-153

   1. Initial program 47.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in lambda2 around inf

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

   8. Applied rewrites 29.4%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(\lambda_2 + {\phi_1}^{2} \cdot \color{blue}{\left(\frac{-1}{8} \cdot \lambda_2 + \frac{1}{384} \cdot \left(\lambda_2 \cdot {\phi_1}^{2}\right)\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 29.6%

       \[\leadsto R \cdot \mathsf{fma}\left(\phi_1 \cdot \phi_1, \mathsf{fma}\left(0.0026041666666666665, \color{blue}{\lambda_2 \cdot \left(\phi_1 \cdot \phi_1\right)}, -0.125 \cdot \lambda_2\right), \lambda_2\right) \]

   if 1.7499999999999999e-153 < phi2 < 1e16

   1. Initial program 44.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

   if 1e16 < phi2

   1. Initial program 48.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 65.7

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

   5. Applied rewrites 65.7%

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

Alternative 9: 58.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-248}:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{elif}\;\phi_2 \leq 1.7 \cdot 10^{-153}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_1 \cdot \phi_1, \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 33000000000000:\\ \;\;\;\;\left(-\phi_1\right) \cdot \left(R - \frac{R \cdot \phi_2}{\phi_1}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \end{array} \]

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 1.7e-248)
   (* (- phi1) (- R (* R (/ phi2 phi1))))
   (if (<= phi2 1.7e-153)
     (* R (fma (* -0.125 lambda2) (* phi1 phi1) lambda2))
     (if (<= phi2 33000000000000.0)
       (* (- phi1) (- R (/ (* R phi2) phi1)))
       (* (fma R (/ (- phi1) phi2) 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 (phi2 <= 1.7e-248) {
		tmp = -phi1 * (R - (R * (phi2 / phi1)));
	} else if (phi2 <= 1.7e-153) {
		tmp = R * fma((-0.125 * lambda2), (phi1 * phi1), lambda2);
	} else if (phi2 <= 33000000000000.0) {
		tmp = -phi1 * (R - ((R * phi2) / phi1));
	} else {
		tmp = fma(R, (-phi1 / phi2), 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 (phi2 <= 1.7e-248)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))));
	elseif (phi2 <= 1.7e-153)
		tmp = Float64(R * fma(Float64(-0.125 * lambda2), Float64(phi1 * phi1), lambda2));
	elseif (phi2 <= 33000000000000.0)
		tmp = Float64(Float64(-phi1) * Float64(R - Float64(Float64(R * phi2) / phi1)));
	else
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * 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, 1.7e-248], N[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.7e-153], N[(R * N[(N[(-0.125 * lambda2), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + lambda2), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 33000000000000.0], N[((-phi1) * N[(R - N[(N[(R * phi2), $MachinePrecision] / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if phi2 < 1.6999999999999999e-248

   1. Initial program 54.0%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 21.2

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

   5. Applied rewrites 21.2%

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

   if 1.6999999999999999e-248 < phi2 < 1.6999999999999999e-153

   1. Initial program 47.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in lambda2 around inf

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

   8. Applied rewrites 29.4%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \color{blue}{\left(\lambda_2 \cdot {\phi_1}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 33.5%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_1 \cdot \color{blue}{\phi_1}, \lambda_2\right) \]

   if 1.6999999999999999e-153 < phi2 < 3.3e13

   1. Initial program 44.4%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 92.8

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

   5. Applied rewrites 92.8%

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

   6. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 41.0

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

   8. Applied rewrites 41.0%

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

   if 3.3e13 < phi2

   1. Initial program 48.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 65.7

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

   5. Applied rewrites 65.7%

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

Alternative 10: 58.7% accurate, 6.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \left(-\phi_1\right) \cdot \left(R - R \cdot \frac{\phi_2}{\phi_1}\right)\\ \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 1.7 \cdot 10^{-153}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_1 \cdot \phi_1, \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 2 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\left(1 - \frac{\phi_1}{\phi_2}\right) \cdot \phi_2\right)\\ \end{array} \end{array} \]

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 (* (- phi1) (- R (* R (/ phi2 phi1))))))
   (if (<= phi2 1.7e-248)
     t_0
     (if (<= phi2 1.7e-153)
       (* R (fma (* -0.125 lambda2) (* phi1 phi1) lambda2))
       (if (<= phi2 2e+24) t_0 (* R (* (- 1.0 (/ 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 t_0 = -phi1 * (R - (R * (phi2 / phi1)));
	double tmp;
	if (phi2 <= 1.7e-248) {
		tmp = t_0;
	} else if (phi2 <= 1.7e-153) {
		tmp = R * fma((-0.125 * lambda2), (phi1 * phi1), lambda2);
	} else if (phi2 <= 2e+24) {
		tmp = t_0;
	} else {
		tmp = R * ((1.0 - (phi1 / phi2)) * phi2);
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(-phi1) * Float64(R - Float64(R * Float64(phi2 / phi1))))
	tmp = 0.0
	if (phi2 <= 1.7e-248)
		tmp = t_0;
	elseif (phi2 <= 1.7e-153)
		tmp = Float64(R * fma(Float64(-0.125 * lambda2), Float64(phi1 * phi1), lambda2));
	elseif (phi2 <= 2e+24)
		tmp = t_0;
	else
		tmp = Float64(R * Float64(Float64(1.0 - Float64(phi1 / phi2)) * 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[((-phi1) * N[(R - N[(R * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[phi2, 1.7e-248], t$95$0, If[LessEqual[phi2, 1.7e-153], N[(R * N[(N[(-0.125 * lambda2), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + lambda2), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 2e+24], t$95$0, N[(R * N[(N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.6999999999999999e-248 or 1.6999999999999999e-153 < phi2 < 2e24

   1. Initial program 52.4%

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

   3. Taylor expanded in phi1 around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. mul-1-neg N/A

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

      6. unsub-neg N/A

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

      7. lower--.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 24.1

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

   5. Applied rewrites 24.1%

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

   if 1.6999999999999999e-248 < phi2 < 1.6999999999999999e-153

   1. Initial program 47.0%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in lambda2 around inf

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

   8. Applied rewrites 29.4%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \color{blue}{\left(\lambda_2 \cdot {\phi_1}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 33.5%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_1 \cdot \color{blue}{\phi_1}, \lambda_2\right) \]

   if 2e24 < phi2

   1. Initial program 48.6%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 67.6

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

   5. Applied rewrites 67.6%

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

Alternative 11: 56.5% accurate, 6.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := R \cdot \left(-\phi_1\right)\\ \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{-248}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_2 \leq 2.2 \cdot 10^{-144}:\\ \;\;\;\;R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_1 \cdot \phi_1, \lambda_2\right)\\ \mathbf{elif}\;\phi_2 \leq 1.45 \cdot 10^{-45}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(R, \frac{-\phi_1}{\phi_2}, R\right) \cdot \phi_2\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* R (- phi1))))
   (if (<= phi2 1.7e-248)
     t_0
     (if (<= phi2 2.2e-144)
       (* R (fma (* -0.125 lambda2) (* phi1 phi1) lambda2))
       (if (<= phi2 1.45e-45) t_0 (* (fma R (/ (- phi1) phi2) 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 t_0 = R * -phi1;
	double tmp;
	if (phi2 <= 1.7e-248) {
		tmp = t_0;
	} else if (phi2 <= 2.2e-144) {
		tmp = R * fma((-0.125 * lambda2), (phi1 * phi1), lambda2);
	} else if (phi2 <= 1.45e-45) {
		tmp = t_0;
	} else {
		tmp = fma(R, (-phi1 / phi2), R) * phi2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(R * Float64(-phi1))
	tmp = 0.0
	if (phi2 <= 1.7e-248)
		tmp = t_0;
	elseif (phi2 <= 2.2e-144)
		tmp = Float64(R * fma(Float64(-0.125 * lambda2), Float64(phi1 * phi1), lambda2));
	elseif (phi2 <= 1.45e-45)
		tmp = t_0;
	else
		tmp = Float64(fma(R, Float64(Float64(-phi1) / phi2), R) * 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[(R * (-phi1)), $MachinePrecision]}, If[LessEqual[phi2, 1.7e-248], t$95$0, If[LessEqual[phi2, 2.2e-144], N[(R * N[(N[(-0.125 * lambda2), $MachinePrecision] * N[(phi1 * phi1), $MachinePrecision] + lambda2), $MachinePrecision]), $MachinePrecision], If[LessEqual[phi2, 1.45e-45], t$95$0, N[(N[(R * N[((-phi1) / phi2), $MachinePrecision] + R), $MachinePrecision] * phi2), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.6999999999999999e-248 or 2.20000000000000006e-144 < phi2 < 1.45e-45

   1. Initial program 54.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   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 26.1

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

   5. Applied rewrites 26.1%

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

   if 1.6999999999999999e-248 < phi2 < 2.20000000000000006e-144

   1. Initial program 47.3%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-cos.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower--.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in lambda2 around inf

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

   8. Applied rewrites 27.2%

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

   9. Taylor expanded in phi1 around 0

      \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \color{blue}{\left(\lambda_2 \cdot {\phi_1}^{2}\right)}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 31.0%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_1 \cdot \color{blue}{\phi_1}, \lambda_2\right) \]

   if 1.45e-45 < phi2

   1. Initial program 45.1%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. mul-1-neg N/A

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

      5. associate-/l* N/A

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

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

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. associate-*r/ N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 12: 53.4% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.7 \cdot 10^{+18}:\\ \;\;\;\;R \cdot \left(-\phi_1\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \phi_2\\ \end{array} \end{array} \]

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 -1.7e+18) (* 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 <= -1.7e+18) {
		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 <= (-1.7d+18)) 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 <= -1.7e+18) {
		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 <= -1.7e+18:
		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 <= -1.7e+18)
		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 <= -1.7e+18)
		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, -1.7e+18], N[(R * (-phi1)), $MachinePrecision], N[(R * phi2), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.7e18

   1. Initial program 45.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \phi_1\right)} \]
   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 69.6

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

   5. Applied rewrites 69.6%

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

   if -1.7e18 < phi1

   1. Initial program 52.7%

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

   3. Taylor expanded in phi2 around inf

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

      1. lower-*.f64 21.6

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

   5. Applied rewrites 21.6%

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

Alternative 13: 32.0% 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 50.9%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 20.3

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

5. Applied rewrites 20.3%

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

Reproduce

herbie shell --seed 2024313 
(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))))))