Equirectangular approximation to distance on a great circle

Percentage Accurate: 60.2% → 91.4%

Time: 10.9s

Alternatives: 13

Speedup: 6.3×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.2% 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: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\phi_1 \leq -1600000:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(t\_0 \cdot \cos \left(0.5 \cdot \phi_1\right), \lambda_2, \left(-\lambda_2\right) \cdot \left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot t\_0, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* phi2 0.5))))
   (if (<= phi1 -1600000.0)
     (*
      (hypot
       (fma
        (* t_0 (cos (* 0.5 phi1)))
        lambda2
        (* (- lambda2) (* (sin (* phi2 0.5)) (sin (* 0.5 phi1)))))
       (- phi1 phi2))
      R)
     (* (hypot (* (- lambda1 lambda2) t_0) phi2) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 * 0.5));
	double tmp;
	if (phi1 <= -1600000.0) {
		tmp = hypot(fma((t_0 * cos((0.5 * phi1))), lambda2, (-lambda2 * (sin((phi2 * 0.5)) * sin((0.5 * phi1))))), (phi1 - phi2)) * R;
	} else {
		tmp = hypot(((lambda1 - lambda2) * t_0), phi2) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 * 0.5))
	tmp = 0.0
	if (phi1 <= -1600000.0)
		tmp = Float64(hypot(fma(Float64(t_0 * cos(Float64(0.5 * phi1))), lambda2, Float64(Float64(-lambda2) * Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1))))), Float64(phi1 - phi2)) * R);
	else
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * t_0), phi2) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1600000.0], N[(N[Sqrt[N[(N[(t$95$0 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * lambda2 + N[((-lambda2) * N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.6e6

   1. Initial program 51.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 87.5%

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

   9. Applied rewrites 87.5%

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

   if -1.6e6 < phi1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1600000:\\ \;\;\;\;\mathsf{hypot}\left(\mathsf{fma}\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \lambda_2, \left(-\lambda_2\right) \cdot \left(\sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right)\right), \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \cos \left(\phi_2 \cdot 0.5\right)\\ \mathbf{if}\;\phi_1 \leq -1600000:\\ \;\;\;\;\mathsf{hypot}\left(\left(t\_0 \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot t\_0, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (cos (* phi2 0.5))))
   (if (<= phi1 -1600000.0)
     (*
      (hypot
       (*
        (-
         (* t_0 (cos (* 0.5 phi1)))
         (* (sin (* phi2 0.5)) (sin (* 0.5 phi1))))
        lambda2)
       (- phi1 phi2))
      R)
     (* (hypot (* (- lambda1 lambda2) t_0) phi2) R))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = cos((phi2 * 0.5));
	double tmp;
	if (phi1 <= -1600000.0) {
		tmp = hypot((((t_0 * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = hypot(((lambda1 - lambda2) * t_0), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = Math.cos((phi2 * 0.5));
	double tmp;
	if (phi1 <= -1600000.0) {
		tmp = Math.hypot((((t_0 * Math.cos((0.5 * phi1))) - (Math.sin((phi2 * 0.5)) * Math.sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = Math.hypot(((lambda1 - lambda2) * t_0), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	t_0 = math.cos((phi2 * 0.5))
	tmp = 0
	if phi1 <= -1600000.0:
		tmp = math.hypot((((t_0 * math.cos((0.5 * phi1))) - (math.sin((phi2 * 0.5)) * math.sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R
	else:
		tmp = math.hypot(((lambda1 - lambda2) * t_0), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos(Float64(phi2 * 0.5))
	tmp = 0.0
	if (phi1 <= -1600000.0)
		tmp = Float64(hypot(Float64(Float64(Float64(t_0 * cos(Float64(0.5 * phi1))) - Float64(sin(Float64(phi2 * 0.5)) * sin(Float64(0.5 * phi1)))) * lambda2), Float64(phi1 - phi2)) * R);
	else
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * t_0), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	t_0 = cos((phi2 * 0.5));
	tmp = 0.0;
	if (phi1 <= -1600000.0)
		tmp = hypot((((t_0 * cos((0.5 * phi1))) - (sin((phi2 * 0.5)) * sin((0.5 * phi1)))) * lambda2), (phi1 - phi2)) * R;
	else
		tmp = hypot(((lambda1 - lambda2) * t_0), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[phi1, -1600000.0], N[(N[Sqrt[N[(N[(N[(t$95$0 * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * t$95$0), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.6e6

   1. Initial program 51.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 87.5%

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

   if -1.6e6 < phi1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1600000:\\ \;\;\;\;\mathsf{hypot}\left(\left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \cos \left(0.5 \cdot \phi_1\right) - \sin \left(\phi_2 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_1\right)\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1600000:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1600000.0)
   (* (hypot (* (cos (* (+ phi2 phi1) 0.5)) lambda2) (- phi1 phi2)) R)
   (* (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1600000.0) {
		tmp = hypot((cos(((phi2 + phi1) * 0.5)) * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1600000.0) {
		tmp = Math.hypot((Math.cos(((phi2 + phi1) * 0.5)) * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1600000.0:
		tmp = math.hypot((math.cos(((phi2 + phi1) * 0.5)) * lambda2), (phi1 - phi2)) * R
	else:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1600000.0)
		tmp = Float64(hypot(Float64(cos(Float64(Float64(phi2 + phi1) * 0.5)) * lambda2), Float64(phi1 - phi2)) * R);
	else
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1600000.0)
		tmp = hypot((cos(((phi2 + phi1) * 0.5)) * lambda2), (phi1 - phi2)) * R;
	else
		tmp = hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1600000.0], N[(N[Sqrt[N[(N[Cos[N[(N[(phi2 + phi1), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.6e6

   1. Initial program 51.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 83.5

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

   5. Applied rewrites 83.5%

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

   if -1.6e6 < phi1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1600000:\\ \;\;\;\;\mathsf{hypot}\left(\cos \left(\left(\phi_2 + \phi_1\right) \cdot 0.5\right) \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.82 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -1.82e-12)
   (* (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi1))) phi1) R)
   (* (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.82e-12) {
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.82e-12) {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi1))), phi1) * R;
	} else {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.82e-12:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi1))), phi1) * R
	else:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.82e-12)
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi1))), phi1) * R);
	else
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.82e-12)
		tmp = hypot(((lambda1 - lambda2) * cos((0.5 * phi1))), phi1) * R;
	else
		tmp = hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -1.82e-12], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi1 ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.82e-12

   1. Initial program 49.5%

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

   3. Taylor expanded in phi2 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower-cos.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 75.6

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

   5. Applied rewrites 75.6%

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

   if -1.82e-12 < phi1

   1. Initial program 64.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 81.9

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

   5. Applied rewrites 81.9%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.82 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_1\right), \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4200000:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4200000.0)
   (* (hypot (* 1.0 lambda2) (- phi1 phi2)) R)
   (* (hypot (* (- lambda1 lambda2) (cos (* phi2 0.5))) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4200000.0) {
		tmp = hypot((1.0 * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4200000.0) {
		tmp = Math.hypot((1.0 * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = Math.hypot(((lambda1 - lambda2) * Math.cos((phi2 * 0.5))), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4200000.0:
		tmp = math.hypot((1.0 * lambda2), (phi1 - phi2)) * R
	else:
		tmp = math.hypot(((lambda1 - lambda2) * math.cos((phi2 * 0.5))), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4200000.0)
		tmp = Float64(hypot(Float64(1.0 * lambda2), Float64(phi1 - phi2)) * R);
	else
		tmp = Float64(hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi2 * 0.5))), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4200000.0)
		tmp = hypot((1.0 * lambda2), (phi1 - phi2)) * R;
	else
		tmp = hypot(((lambda1 - lambda2) * cos((phi2 * 0.5))), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4200000.0], N[(N[Sqrt[N[(1.0 * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi2 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.2e6

   1. Initial program 51.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 72.8%

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

   11. Taylor expanded in phi1 around 0

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

   13. Applied rewrites 77.5%

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \lambda_2, \phi_1 - \phi_2\right) \]

   if -4.2e6 < phi1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 82.1

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

   5. Applied rewrites 82.1%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4200000:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_2 \cdot 0.5\right), \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.2% accurate, 2.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4200000:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -4200000.0)
   (* (hypot (* 1.0 lambda2) (- phi1 phi2)) R)
   (* (hypot (- lambda1 lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4200000.0) {
		tmp = hypot((1.0 * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -4200000.0) {
		tmp = Math.hypot((1.0 * lambda2), (phi1 - phi2)) * R;
	} else {
		tmp = Math.hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -4200000.0:
		tmp = math.hypot((1.0 * lambda2), (phi1 - phi2)) * R
	else:
		tmp = math.hypot((lambda1 - lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -4200000.0)
		tmp = Float64(hypot(Float64(1.0 * lambda2), Float64(phi1 - phi2)) * R);
	else
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -4200000.0)
		tmp = hypot((1.0 * lambda2), (phi1 - phi2)) * R;
	else
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -4200000.0], N[(N[Sqrt[N[(1.0 * lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -4.2e6

   1. Initial program 51.5%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 83.5

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

   5. Applied rewrites 83.5%

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

   7. Applied rewrites 87.5%

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

   8. Taylor expanded in phi2 around 0

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

   10. Applied rewrites 72.8%

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

   11. Taylor expanded in phi1 around 0

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

   13. Applied rewrites 77.5%

       \[\leadsto R \cdot \mathsf{hypot}\left(1 \cdot \lambda_2, \phi_1 - \phi_2\right) \]

   if -4.2e6 < phi1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 75.1%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -4200000:\\ \;\;\;\;\mathsf{hypot}\left(1 \cdot \lambda_2, \phi_1 - \phi_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.2% accurate, 2.4× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -520000000:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= phi1 -520000000.0)
   (* (- phi2 phi1) R)
   (* (hypot (- lambda1 lambda2) phi2) R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -520000000.0) {
		tmp = (phi2 - phi1) * R;
	} else {
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -520000000.0) {
		tmp = (phi2 - phi1) * R;
	} else {
		tmp = Math.hypot((lambda1 - lambda2), phi2) * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -520000000.0:
		tmp = (phi2 - phi1) * R
	else:
		tmp = math.hypot((lambda1 - lambda2), phi2) * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -520000000.0)
		tmp = Float64(Float64(phi2 - phi1) * R);
	else
		tmp = Float64(hypot(Float64(lambda1 - lambda2), phi2) * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -520000000.0)
		tmp = (phi2 - phi1) * R;
	else
		tmp = hypot((lambda1 - lambda2), phi2) * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[phi1, -520000000.0], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], N[(N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + phi2 ^ 2], $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -5.2e8

   1. Initial program 51.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   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 72.5

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

   5. Applied rewrites 72.5%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 72.5%

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

   if -5.2e8 < phi1

   1. Initial program 63.0%

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

   3. Taylor expanded in phi1 around 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. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lower--.f64 82.1

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

   5. Applied rewrites 82.1%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 75.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -520000000:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.2% accurate, 6.3× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.2 \cdot 10^{+208}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, \left(\phi_2 \cdot \phi_2\right) \cdot \lambda_2, -0.125 \cdot \lambda_2\right), \phi_2 \cdot \phi_2, \lambda_2\right) \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (if (<= lambda2 9.2e+208)
   (* (- phi2 phi1) R)
   (*
    (fma
     (fma 0.0026041666666666665 (* (* phi2 phi2) lambda2) (* -0.125 lambda2))
     (* phi2 phi2)
     lambda2)
    R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 9.2e+208) {
		tmp = (phi2 - phi1) * R;
	} else {
		tmp = fma(fma(0.0026041666666666665, ((phi2 * phi2) * lambda2), (-0.125 * lambda2)), (phi2 * phi2), lambda2) * R;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 9.2e+208)
		tmp = Float64(Float64(phi2 - phi1) * R);
	else
		tmp = Float64(fma(fma(0.0026041666666666665, Float64(Float64(phi2 * phi2) * lambda2), Float64(-0.125 * lambda2)), Float64(phi2 * phi2), lambda2) * R);
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := If[LessEqual[lambda2, 9.2e+208], N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], N[(N[(N[(0.0026041666666666665 * N[(N[(phi2 * phi2), $MachinePrecision] * lambda2), $MachinePrecision] + N[(-0.125 * lambda2), $MachinePrecision]), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + lambda2), $MachinePrecision] * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 9.20000000000000002e208

   1. Initial program 61.5%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   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 29.3

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

   5. Applied rewrites 29.3%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 31.4%

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

   if 9.20000000000000002e208 < lambda2

   1. Initial program 45.0%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 95.6

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

   5. Applied rewrites 95.6%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 77.9%

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

   9. Taylor expanded in lambda2 around inf

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

   11. Applied rewrites 60.5%

       \[\leadsto R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_2\right) \]

   12. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(\lambda_2 + {\phi_2}^{2} \cdot \left(\frac{-1}{8} \cdot \lambda_2 + \color{blue}{\frac{1}{384} \cdot \left(\lambda_2 \cdot {\phi_2}^{2}\right)}\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 66.7%

       \[\leadsto R \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, \left(\phi_2 \cdot \phi_2\right) \cdot \lambda_2, -0.125 \cdot \lambda_2\right), \phi_2 \cdot \phi_2, \lambda_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 9.2 \cdot 10^{+208}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.0026041666666666665, \left(\phi_2 \cdot \phi_2\right) \cdot \lambda_2, -0.125 \cdot \lambda_2\right), \phi_2 \cdot \phi_2, \lambda_2\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 59.1% accurate, 7.5× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-51}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_2 \cdot \phi_2, \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(R - \frac{\phi_1}{\phi_2} \cdot R\right) \cdot \phi_2\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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)
   (* (- phi2 phi1) R)
   (if (<= phi1 -1.05e-139)
     (* (fma (* -0.125 lambda2) (* phi2 phi2) lambda2) R)
     (* (- R (* (/ phi1 phi2) R)) phi2))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.1e-51) {
		tmp = (phi2 - phi1) * R;
	} else if (phi1 <= -1.05e-139) {
		tmp = fma((-0.125 * lambda2), (phi2 * phi2), lambda2) * R;
	} else {
		tmp = (R - ((phi1 / phi2) * R)) * phi2;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.1e-51)
		tmp = Float64(Float64(phi2 - phi1) * R);
	elseif (phi1 <= -1.05e-139)
		tmp = Float64(fma(Float64(-0.125 * lambda2), Float64(phi2 * phi2), lambda2) * R);
	else
		tmp = Float64(Float64(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.
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[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision], If[LessEqual[phi1, -1.05e-139], N[(N[(N[(-0.125 * lambda2), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + lambda2), $MachinePrecision] * R), $MachinePrecision], N[(N[(R - N[(N[(phi1 / phi2), $MachinePrecision] * R), $MachinePrecision]), $MachinePrecision] * phi2), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if phi1 < -1.1e-51

   1. Initial program 53.7%

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

   3. Taylor expanded in 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 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 65.6%

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

   if -1.1e-51 < phi1 < -1.05000000000000004e-139

   1. Initial program 68.4%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in lambda2 around inf

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

   11. Applied rewrites 30.5%

       \[\leadsto R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_2\right) \]

   12. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \left(\lambda_2 \cdot \color{blue}{{\phi_2}^{2}}\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 31.1%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_2 \cdot \phi_2, \lambda_2\right) \]

   if -1.05000000000000004e-139 < phi1

   1. Initial program 62.3%

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

   3. Taylor expanded in phi2 around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 N/A

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

      6. associate-/l* N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 19.2

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

   5. Applied rewrites 19.2%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-51}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -1.05 \cdot 10^{-139}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_2 \cdot \phi_2, \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(R - \frac{\phi_1}{\phi_2} \cdot R\right) \cdot \phi_2\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.9% accurate, 8.2× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} t_0 := \left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-51}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;\phi_1 \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_2 \cdot \phi_2, \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R lambda1 lambda2 phi1 phi2)
 :precision binary64
 (let* ((t_0 (* (- phi2 phi1) R)))
   (if (<= phi1 -1.1e-51)
     t_0
     (if (<= phi1 -3.8e-140)
       (* (fma (* -0.125 lambda2) (* phi2 phi2) lambda2) R)
       t_0))))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double t_0 = (phi2 - phi1) * R;
	double tmp;
	if (phi1 <= -1.1e-51) {
		tmp = t_0;
	} else if (phi1 <= -3.8e-140) {
		tmp = fma((-0.125 * lambda2), (phi2 * phi2), lambda2) * R;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	t_0 = Float64(Float64(phi2 - phi1) * R)
	tmp = 0.0
	if (phi1 <= -1.1e-51)
		tmp = t_0;
	elseif (phi1 <= -3.8e-140)
		tmp = Float64(fma(Float64(-0.125 * lambda2), Float64(phi2 * phi2), lambda2) * R);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R_, lambda1_, lambda2_, phi1_, phi2_] := Block[{t$95$0 = N[(N[(phi2 - phi1), $MachinePrecision] * R), $MachinePrecision]}, If[LessEqual[phi1, -1.1e-51], t$95$0, If[LessEqual[phi1, -3.8e-140], N[(N[(N[(-0.125 * lambda2), $MachinePrecision] * N[(phi2 * phi2), $MachinePrecision] + lambda2), $MachinePrecision] * R), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.1e-51 or -3.79999999999999998e-140 < phi1

   1. Initial program 59.6%

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

   3. Taylor expanded in phi1 around -inf

      \[\leadsto R \cdot \color{blue}{\left(-1 \cdot \left(\phi_1 \cdot \left(1 + -1 \cdot \frac{\phi_2}{\phi_1}\right)\right)\right)} \]
   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 31.1

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

   5. Applied rewrites 31.1%

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

   6. Taylor expanded in phi2 around 0

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

   8. Applied rewrites 33.3%

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

   if -1.1e-51 < phi1 < -3.79999999999999998e-140

   1. Initial program 68.4%

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

   3. Taylor expanded in lambda1 around 0

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

      1. unpow2 N/A

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

      2. unpow2 N/A

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

      3. unswap-sqr N/A

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

      4. unpow2 N/A

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

      5. lower-hypot.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-cos.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 N/A

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

      13. lower--.f64 59.1

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

   5. Applied rewrites 59.1%

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

   6. Taylor expanded in phi1 around 0

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

   8. Applied rewrites 59.1%

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

   9. Taylor expanded in lambda2 around inf

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

   11. Applied rewrites 30.5%

       \[\leadsto R \cdot \left(\cos \left(\phi_2 \cdot 0.5\right) \cdot \lambda_2\right) \]

   12. Taylor expanded in phi2 around 0

       \[\leadsto R \cdot \left(\lambda_2 + \frac{-1}{8} \cdot \left(\lambda_2 \cdot \color{blue}{{\phi_2}^{2}}\right)\right) \]
   13. Step-by-step derivation

   14. Applied rewrites 31.1%

       \[\leadsto R \cdot \mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_2 \cdot \phi_2, \lambda_2\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.1 \cdot 10^{-51}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \mathbf{elif}\;\phi_1 \leq -3.8 \cdot 10^{-140}:\\ \;\;\;\;\mathsf{fma}\left(-0.125 \cdot \lambda_2, \phi_2 \cdot \phi_2, \lambda_2\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\left(\phi_2 - \phi_1\right) \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 52.8% accurate, 19.9× speedup

Math

\[\begin{array}{l} [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\\\ [R, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R, lambda1, lambda2, phi1, phi2])\\ \\ \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{+32}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \end{array} \]

FPCore

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 -2.6e+32) (* (- phi1) R) (* phi2 R)))

C

assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
assert(R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.6e+32) {
		tmp = -phi1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Fortran

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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 <= (-2.6d+32)) then
        tmp = -phi1 * r
    else
        tmp = phi2 * r
    end if
    code = tmp
end function

Java

assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
assert R < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -2.6e+32) {
		tmp = -phi1 * R;
	} else {
		tmp = phi2 * R;
	}
	return tmp;
}

Python

[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
[R, lambda1, lambda2, phi1, phi2] = sort([R, lambda1, lambda2, phi1, phi2])
def code(R, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -2.6e+32:
		tmp = -phi1 * R
	else:
		tmp = phi2 * R
	return tmp

Julia

R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
R, lambda1, lambda2, phi1, phi2 = sort([R, lambda1, lambda2, phi1, phi2])
function code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -2.6e+32)
		tmp = Float64(Float64(-phi1) * R);
	else
		tmp = Float64(phi2 * R);
	end
	return tmp
end

MATLAB

R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
R, lambda1, lambda2, phi1, phi2 = num2cell(sort([R, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -2.6e+32)
		tmp = -phi1 * R;
	else
		tmp = phi2 * R;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: R, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
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, -2.6e+32], N[((-phi1) * R), $MachinePrecision], N[(phi2 * R), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if phi1 < -2.6000000000000002e32

   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. 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 60.6

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

   5. Applied rewrites 60.6%

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

   if -2.6000000000000002e32 < phi1

   1. Initial program 63.7%

      \[R \cdot \sqrt{\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) \cdot \left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right)\right) + \left(\phi_1 - \phi_2\right) \cdot \left(\phi_1 - \phi_2\right)} \]
   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 20.0

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

   5. Applied rewrites 20.0%

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

4. Final simplification 29.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -2.6 \cdot 10^{+32}:\\ \;\;\;\;\left(-\phi_1\right) \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.7% accurate, 31.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in phi1 around -inf

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

   1. 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 29.2

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

5. Applied rewrites 29.2%

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

6. Taylor expanded in phi2 around 0

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

8. Applied rewrites 31.2%

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

9. Final simplification 31.2%

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

Alternative 13: 32.0% accurate, 46.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.2%

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

3. Taylor expanded in phi2 around inf

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

   1. lower-*.f64 19.4

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

5. Applied rewrites 19.4%

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

6. Final simplification 19.4%

   \[\leadsto \phi_2 \cdot R \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024278 
(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))))))