Equirectangular approximation to distance on a great circle

Percentage Accurate: 59.6% → 99.2%

Time: 18.9s

Alternatives: 16

Speedup: 3.0×

• 31.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 59.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\phi_1 \cdot 0.5\right)\right)\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R\_m}\right)}^{2} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (pow
   (*
    (sqrt
     (hypot
      (*
       (- lambda1 lambda2)
       (-
        (* (log1p (expm1 (cos (* phi1 0.5)))) (cos (* 0.5 phi2)))
        (* (sin (* phi1 0.5)) (sin (* 0.5 phi2)))))
      (- phi1 phi2)))
    (sqrt R_m))
   2.0)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * pow((sqrt(hypot(((lambda1 - lambda2) * ((log1p(expm1(cos((phi1 * 0.5)))) * cos((0.5 * phi2))) - (sin((phi1 * 0.5)) * sin((0.5 * phi2))))), (phi1 - phi2))) * sqrt(R_m)), 2.0);
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * Math.pow((Math.sqrt(Math.hypot(((lambda1 - lambda2) * ((Math.log1p(Math.expm1(Math.cos((phi1 * 0.5)))) * Math.cos((0.5 * phi2))) - (Math.sin((phi1 * 0.5)) * Math.sin((0.5 * phi2))))), (phi1 - phi2))) * Math.sqrt(R_m)), 2.0);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * math.pow((math.sqrt(math.hypot(((lambda1 - lambda2) * ((math.log1p(math.expm1(math.cos((phi1 * 0.5)))) * math.cos((0.5 * phi2))) - (math.sin((phi1 * 0.5)) * math.sin((0.5 * phi2))))), (phi1 - phi2))) * math.sqrt(R_m)), 2.0)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * (Float64(sqrt(hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(log1p(expm1(cos(Float64(phi1 * 0.5)))) * cos(Float64(0.5 * phi2))) - Float64(sin(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2))))), Float64(phi1 - phi2))) * sqrt(R_m)) ^ 2.0))
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[Power[N[(N[Sqrt[N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Log[1 + N[(Exp[N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision] * N[Sqrt[R$95$m], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. hypot-define 96.7%

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

3. Simplified 96.7%

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

   1. add-sqr-sqrt 45.7%

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

   2. pow2 45.7%

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

   3. div-inv 45.7%

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

   4. metadata-eval 45.7%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

6. Applied egg-rr 45.7%

   \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 45.7%

      \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right) \cdot R}}\right)}^{2} \]

   2. sqrt-prod 45.2%

      \[\leadsto {\color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}}^{2} \]

8. Applied egg-rr 45.2%

   \[\leadsto {\color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}}^{2} \]
9. Step-by-step derivation

   1. *-commutative 45.2%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   2. distribute-lft-in 45.2%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1 + 0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   3. cos-sum 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   4. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   5. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   6. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   7. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

10. Applied egg-rr 46.4%

    \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]
11. Step-by-step derivation

    1. log1p-expm1-u 46.4%

       \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\phi_1 \cdot 0.5\right)\right)\right)} \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

12. Applied egg-rr 46.4%

    \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\phi_1 \cdot 0.5\right)\right)\right)} \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

13. Final simplification 46.4%

    \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos \left(\phi_1 \cdot 0.5\right)\right)\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]
14. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot {\left(\sqrt{R\_m} \cdot \sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (pow
   (*
    (sqrt R_m)
    (sqrt
     (hypot
      (*
       (- lambda1 lambda2)
       (-
        (* (cos (* 0.5 phi2)) (cos (* phi1 0.5)))
        (* (sin (* phi1 0.5)) (sin (* 0.5 phi2)))))
      (- phi1 phi2))))
   2.0)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * pow((sqrt(R_m) * sqrt(hypot(((lambda1 - lambda2) * ((cos((0.5 * phi2)) * cos((phi1 * 0.5))) - (sin((phi1 * 0.5)) * sin((0.5 * phi2))))), (phi1 - phi2)))), 2.0);
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * Math.pow((Math.sqrt(R_m) * Math.sqrt(Math.hypot(((lambda1 - lambda2) * ((Math.cos((0.5 * phi2)) * Math.cos((phi1 * 0.5))) - (Math.sin((phi1 * 0.5)) * Math.sin((0.5 * phi2))))), (phi1 - phi2)))), 2.0);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * math.pow((math.sqrt(R_m) * math.sqrt(math.hypot(((lambda1 - lambda2) * ((math.cos((0.5 * phi2)) * math.cos((phi1 * 0.5))) - (math.sin((phi1 * 0.5)) * math.sin((0.5 * phi2))))), (phi1 - phi2)))), 2.0)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * (Float64(sqrt(R_m) * sqrt(hypot(Float64(Float64(lambda1 - lambda2) * Float64(Float64(cos(Float64(0.5 * phi2)) * cos(Float64(phi1 * 0.5))) - Float64(sin(Float64(phi1 * 0.5)) * sin(Float64(0.5 * phi2))))), Float64(phi1 - phi2)))) ^ 2.0))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * ((sqrt(R_m) * sqrt(hypot(((lambda1 - lambda2) * ((cos((0.5 * phi2)) * cos((phi1 * 0.5))) - (sin((phi1 * 0.5)) * sin((0.5 * phi2))))), (phi1 - phi2)))) ^ 2.0);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[Power[N[(N[Sqrt[R$95$m], $MachinePrecision] * N[Sqrt[N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[(N[(N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(N[Sin[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision] * N[Sin[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. hypot-define 96.7%

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

3. Simplified 96.7%

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

   1. add-sqr-sqrt 45.7%

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

   2. pow2 45.7%

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

   3. div-inv 45.7%

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

   4. metadata-eval 45.7%

      \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

6. Applied egg-rr 45.7%

   \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 45.7%

      \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right) \cdot R}}\right)}^{2} \]

   2. sqrt-prod 45.2%

      \[\leadsto {\color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}}^{2} \]

8. Applied egg-rr 45.2%

   \[\leadsto {\color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}}^{2} \]
9. Step-by-step derivation

   1. *-commutative 45.2%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   2. distribute-lft-in 45.2%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1 + 0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   3. cos-sum 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   4. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   5. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   6. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   7. *-commutative 46.4%

      \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

10. Applied egg-rr 46.4%

    \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

11. Final simplification 46.4%

    \[\leadsto {\left(\sqrt{R} \cdot \sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(0.5 \cdot \phi_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)}\right)}^{2} \]
12. Add Preprocessing

Alternative 3: 95.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.2 \cdot 10^{-33}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 5.2e-33)
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))
    (* R_m (hypot (* (- lambda1 lambda2) (cos (* 0.5 phi2))) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.2e-33) {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 5.2e-33) {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((0.5 * phi2))), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 5.2e-33:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot(((lambda1 - lambda2) * math.cos((0.5 * phi2))), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 5.2e-33)
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(0.5 * phi2))), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 5.2e-33)
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R_m * hypot(((lambda1 - lambda2) * cos((0.5 * phi2))), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 5.2e-33], N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(0.5 * phi2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 5.19999999999999988e-33

   1. Initial program 65.2%

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

      1. hypot-define 98.1%

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

   3. Simplified 98.1%

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

   5. Taylor expanded in phi2 around 0 93.1%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 93.1%

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

   7. Simplified 93.1%

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

   if 5.19999999999999988e-33 < phi2

   1. Initial program 57.9%

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

      1. hypot-define 93.0%

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

   3. Simplified 93.0%

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

   5. Taylor expanded in phi1 around 0 91.9%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_2\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 91.9%

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

   7. Simplified 91.9%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 5.2 \cdot 10^{-33}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(0.5 \cdot \phi_2\right), \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{+100}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda1 -3e+100)
    (* R_m (hypot (* lambda1 (cos (* phi1 0.5))) (- phi1 phi2)))
    (* R_m (hypot (- lambda1 lambda2) (- phi1 phi2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3e+100) {
		tmp = R_m * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda1 <= -3e+100) {
		tmp = R_m * Math.hypot((lambda1 * Math.cos((phi1 * 0.5))), (phi1 - phi2));
	} else {
		tmp = R_m * Math.hypot((lambda1 - lambda2), (phi1 - phi2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda1 <= -3e+100:
		tmp = R_m * math.hypot((lambda1 * math.cos((phi1 * 0.5))), (phi1 - phi2))
	else:
		tmp = R_m * math.hypot((lambda1 - lambda2), (phi1 - phi2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda1 <= -3e+100)
		tmp = Float64(R_m * hypot(Float64(lambda1 * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2)));
	else
		tmp = Float64(R_m * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda1 <= -3e+100)
		tmp = R_m * hypot((lambda1 * cos((phi1 * 0.5))), (phi1 - phi2));
	else
		tmp = R_m * hypot((lambda1 - lambda2), (phi1 - phi2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda1, -3e+100], N[(R$95$m * N[Sqrt[N[(lambda1 * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda1 < -2.99999999999999985e100

   1. Initial program 56.3%

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

      1. hypot-define 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in phi2 around 0 79.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 79.3%

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

   7. Simplified 79.3%

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

   8. Taylor expanded in lambda1 around inf 79.3%

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

   if -2.99999999999999985e100 < lambda1

   1. Initial program 64.3%

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

      1. hypot-define 96.6%

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

   3. Simplified 96.6%

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

   5. Taylor expanded in phi2 around 0 91.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in phi1 around 0 88.3%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_1 \leq -3 \cdot 10^{+100}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (*
   R_m
   (hypot (* (- lambda1 lambda2) (cos (/ (+ phi1 phi2) 2.0))) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot(((lambda1 - lambda2) * Math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot(((lambda1 - lambda2) * math.cos(((phi1 + phi2) / 2.0))), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(Float64(phi1 + phi2) / 2.0))), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot(((lambda1 - lambda2) * cos(((phi1 + phi2) / 2.0))), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(N[(phi1 + phi2), $MachinePrecision] / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. hypot-define 96.7%

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

3. Simplified 96.7%

   \[\leadsto \color{blue}{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\frac{\phi_1 + \phi_2}{2}\right), \phi_1 - \phi_2\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 6: 90.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (* R_m (hypot (* (- lambda1 lambda2) (cos (* phi1 0.5))) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot(((lambda1 - lambda2) * Math.cos((phi1 * 0.5))), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot(((lambda1 - lambda2) * math.cos((phi1 * 0.5))), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(Float64(lambda1 - lambda2) * cos(Float64(phi1 * 0.5))), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot(((lambda1 - lambda2) * cos((phi1 * 0.5))), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[Sqrt[N[(N[(lambda1 - lambda2), $MachinePrecision] * N[Cos[N[(phi1 * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. hypot-define 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in phi2 around 0 89.9%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 89.9%

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

7. Simplified 89.9%

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

8. Final simplification 89.9%

   \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\phi_1 \cdot 0.5\right), \phi_1 - \phi_2\right) \]
9. Add Preprocessing

Alternative 7: 80.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{+50}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi1 -1.55e+50)
    (* phi1 (- (* R_m (/ phi2 phi1)) R_m))
    (* R_m (hypot phi2 (- lambda1 lambda2))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.55e+50) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi1 <= -1.55e+50) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = R_m * Math.hypot(phi2, (lambda1 - lambda2));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi1 <= -1.55e+50:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m)
	else:
		tmp = R_m * math.hypot(phi2, (lambda1 - lambda2))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi1 <= -1.55e+50)
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - R_m));
	else
		tmp = Float64(R_m * hypot(phi2, Float64(lambda1 - lambda2)));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi1 <= -1.55e+50)
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	else
		tmp = R_m * hypot(phi2, (lambda1 - lambda2));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi1, -1.55e+50], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[Sqrt[phi2 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi1 < -1.55000000000000001e50

   1. Initial program 57.3%

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

      1. hypot-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in phi1 around -inf 65.2%

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

      1. mul-1-neg 65.2%

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

      2. distribute-rgt-neg-in 65.2%

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

      3. mul-1-neg 65.2%

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

      4. unsub-neg 65.2%

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

      5. associate-/l* 70.6%

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

   7. Simplified 70.6%

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

   if -1.55000000000000001e50 < phi1

   1. Initial program 64.8%

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

      1. hypot-define 97.6%

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

   3. Simplified 97.6%

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

   5. Taylor expanded in phi2 around 0 89.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 89.0%

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

   7. Simplified 89.0%

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

   8. Taylor expanded in phi1 around 0 51.4%

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

      1. unpow2 51.4%

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

      2. unpow2 51.4%

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

      3. hypot-define 71.9%

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

   10. Simplified 71.9%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_1 \leq -1.55 \cdot 10^{+50}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;R \cdot \mathsf{hypot}\left(\phi_2, \lambda_1 - \lambda_2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.7 \cdot 10^{+68}:\\ \;\;\;\;R\_m \cdot \mathsf{hypot}\left(\phi_1, \lambda_1 - \lambda_2\right)\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 1.7e+68)
    (* R_m (hypot phi1 (- lambda1 lambda2)))
    (* R_m (* phi2 (- 1.0 (/ phi1 phi2)))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.7e+68) {
		tmp = R_m * hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.7e+68) {
		tmp = R_m * Math.hypot(phi1, (lambda1 - lambda2));
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.7e+68:
		tmp = R_m * math.hypot(phi1, (lambda1 - lambda2))
	else:
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.7e+68)
		tmp = Float64(R_m * hypot(phi1, Float64(lambda1 - lambda2)));
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.7e+68)
		tmp = R_m * hypot(phi1, (lambda1 - lambda2));
	else
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 1.7e+68], N[(R$95$m * N[Sqrt[phi1 ^ 2 + N[(lambda1 - lambda2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if phi2 < 1.70000000000000008e68

   1. Initial program 65.8%

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

      1. hypot-define 97.8%

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

   3. Simplified 97.8%

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

   5. Taylor expanded in phi1 around 0 85.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_2\right) + -0.5 \cdot \left(\phi_1 \cdot \sin \left(0.5 \cdot \phi_2\right)\right)\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. associate-*r* 85.6%

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

   7. Simplified 85.6%

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

   8. Taylor expanded in phi2 around 0 52.8%

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

      1. unpow2 52.8%

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

      2. unpow2 52.8%

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

      3. hypot-define 70.0%

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

   10. Simplified 70.0%

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

   if 1.70000000000000008e68 < phi2

   1. Initial program 52.8%

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

      1. hypot-define 92.4%

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

   3. Simplified 92.4%

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

   5. Taylor expanded in phi2 around inf 68.4%

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

      1. mul-1-neg 68.4%

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

      2. unsub-neg 68.4%

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

   7. Simplified 68.4%

      \[\leadsto R \cdot \color{blue}{\left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 85.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(R\_m \cdot \mathsf{hypot}\left(\lambda_1 - \lambda_2, \phi_1 - \phi_2\right)\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* R_m (hypot (- lambda1 lambda2) (- phi1 phi2)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * hypot((lambda1 - lambda2), (phi1 - phi2)));
}

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (R_m * Math.hypot((lambda1 - lambda2), (phi1 - phi2)));
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (R_m * math.hypot((lambda1 - lambda2), (phi1 - phi2)))

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(R_m * hypot(Float64(lambda1 - lambda2), Float64(phi1 - phi2))))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (R_m * hypot((lambda1 - lambda2), (phi1 - phi2)));
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(R$95$m * N[Sqrt[N[(lambda1 - lambda2), $MachinePrecision] ^ 2 + N[(phi1 - phi2), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. hypot-define 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in phi2 around 0 89.9%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 89.9%

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

7. Simplified 89.9%

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

8. Taylor expanded in phi1 around 0 85.3%

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

Alternative 10: 55.2% accurate, 17.3× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-175}:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\phi_2 \leq 1.38 \cdot 10^{-61}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \mathbf{else}:\\ \;\;\;\;R\_m \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 1.9e-175)
    (* phi1 (- R_m))
    (if (<= phi2 1.38e-61)
      (* lambda2 R_m)
      (* R_m (* phi2 (- 1.0 (/ phi1 phi2))))))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9e-175) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 1.38e-61) {
		tmp = lambda2 * R_m;
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.9d-175) then
        tmp = phi1 * -r_m
    else if (phi2 <= 1.38d-61) then
        tmp = lambda2 * r_m
    else
        tmp = r_m * (phi2 * (1.0d0 - (phi1 / phi2)))
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.9e-175) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 1.38e-61) {
		tmp = lambda2 * R_m;
	} else {
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.9e-175:
		tmp = phi1 * -R_m
	elif phi2 <= 1.38e-61:
		tmp = lambda2 * R_m
	else:
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)))
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.9e-175)
		tmp = Float64(phi1 * Float64(-R_m));
	elseif (phi2 <= 1.38e-61)
		tmp = Float64(lambda2 * R_m);
	else
		tmp = Float64(R_m * Float64(phi2 * Float64(1.0 - Float64(phi1 / phi2))));
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.9e-175)
		tmp = phi1 * -R_m;
	elseif (phi2 <= 1.38e-61)
		tmp = lambda2 * R_m;
	else
		tmp = R_m * (phi2 * (1.0 - (phi1 / phi2)));
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 1.9e-175], N[(phi1 * (-R$95$m)), $MachinePrecision], If[LessEqual[phi2, 1.38e-61], N[(lambda2 * R$95$m), $MachinePrecision], N[(R$95$m * N[(phi2 * N[(1.0 - N[(phi1 / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.9e-175

   1. Initial program 65.4%

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

      1. hypot-define 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in phi1 around -inf 20.6%

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

      1. mul-1-neg 20.6%

         \[\leadsto \color{blue}{-R \cdot \phi_1} \]

      2. distribute-rgt-neg-in 20.6%

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

   7. Simplified 20.6%

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

   if 1.9e-175 < phi2 < 1.37999999999999992e-61

   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. Step-by-step derivation

      1. hypot-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in phi2 around 0 99.8%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 99.8%

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

   7. Simplified 99.8%

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

   8. Taylor expanded in lambda2 around inf 33.7%

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

   9. Taylor expanded in phi1 around 0 42.7%

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

   if 1.37999999999999992e-61 < phi2

   1. Initial program 58.2%

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

      1. hypot-define 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in phi2 around inf 58.5%

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

      1. mul-1-neg 58.5%

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

      2. unsub-neg 58.5%

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

   7. Simplified 58.5%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.9 \cdot 10^{-175}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 1.38 \cdot 10^{-61}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;R \cdot \left(\phi_2 \cdot \left(1 - \frac{\phi_1}{\phi_2}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.4% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.9 \cdot 10^{+229}:\\ \;\;\;\;\phi_1 \cdot \left(R\_m \cdot \frac{\phi_2}{\phi_1} - R\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 2.9e+229)
    (* phi1 (- (* R_m (/ phi2 phi1)) R_m))
    (* lambda2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.9e+229) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 2.9d+229) then
        tmp = phi1 * ((r_m * (phi2 / phi1)) - r_m)
    else
        tmp = lambda2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.9e+229) {
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.9e+229:
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m)
	else:
		tmp = lambda2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.9e+229)
		tmp = Float64(phi1 * Float64(Float64(R_m * Float64(phi2 / phi1)) - R_m));
	else
		tmp = Float64(lambda2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2.9e+229)
		tmp = phi1 * ((R_m * (phi2 / phi1)) - R_m);
	else
		tmp = lambda2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 2.9e+229], N[(phi1 * N[(N[(R$95$m * N[(phi2 / phi1), $MachinePrecision]), $MachinePrecision] - R$95$m), $MachinePrecision]), $MachinePrecision], N[(lambda2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.89999999999999981e229

   1. Initial program 64.4%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi1 around -inf 31.4%

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

      1. mul-1-neg 31.4%

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

      2. distribute-rgt-neg-in 31.4%

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

      3. mul-1-neg 31.4%

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

      4. unsub-neg 31.4%

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

      5. associate-/l* 31.0%

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

   7. Simplified 31.0%

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

   if 2.89999999999999981e229 < lambda2

   1. Initial program 32.9%

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

      1. hypot-define 83.9%

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

   3. Simplified 83.9%

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

   5. Taylor expanded in phi2 around 0 76.0%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 76.0%

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

   7. Simplified 76.0%

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

   8. Taylor expanded in lambda2 around inf 54.6%

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

   9. Taylor expanded in phi1 around 0 68.9%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.9 \cdot 10^{+229}:\\ \;\;\;\;\phi_1 \cdot \left(R \cdot \frac{\phi_2}{\phi_1} - R\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.8% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.05 \cdot 10^{+165}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \frac{\phi_1 \cdot R\_m}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 2.05e+165)
    (* phi2 (- R_m (/ (* phi1 R_m) phi2)))
    (* lambda2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.05e+165) {
		tmp = phi2 * (R_m - ((phi1 * R_m) / phi2));
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 2.05d+165) then
        tmp = phi2 * (r_m - ((phi1 * r_m) / phi2))
    else
        tmp = lambda2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 2.05e+165) {
		tmp = phi2 * (R_m - ((phi1 * R_m) / phi2));
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 2.05e+165:
		tmp = phi2 * (R_m - ((phi1 * R_m) / phi2))
	else:
		tmp = lambda2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 2.05e+165)
		tmp = Float64(phi2 * Float64(R_m - Float64(Float64(phi1 * R_m) / phi2)));
	else
		tmp = Float64(lambda2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 2.05e+165)
		tmp = phi2 * (R_m - ((phi1 * R_m) / phi2));
	else
		tmp = lambda2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 2.05e+165], N[(phi2 * N[(R$95$m - N[(N[(phi1 * R$95$m), $MachinePrecision] / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(lambda2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 2.0500000000000001e165

   1. Initial program 65.7%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

      1. add-sqr-sqrt 44.6%

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

      2. pow2 44.6%

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

      3. div-inv 44.6%

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

      4. metadata-eval 44.6%

         \[\leadsto {\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot \color{blue}{0.5}\right), \phi_1 - \phi_2\right)}\right)}^{2} \]

   6. Applied egg-rr 44.6%

      \[\leadsto \color{blue}{{\left(\sqrt{R \cdot \mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)}\right)}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 44.6%

         \[\leadsto {\left(\sqrt{\color{blue}{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right) \cdot R}}\right)}^{2} \]

      2. sqrt-prod 44.0%

         \[\leadsto {\color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}}^{2} \]

   8. Applied egg-rr 44.0%

      \[\leadsto {\color{blue}{\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \left(\left(\phi_1 + \phi_2\right) \cdot 0.5\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}}^{2} \]
   9. Step-by-step derivation

      1. *-commutative 44.0%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \left(\phi_1 + \phi_2\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

      2. distribute-lft-in 44.0%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \cos \color{blue}{\left(0.5 \cdot \phi_1 + 0.5 \cdot \phi_2\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

      3. cos-sum 44.8%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(0.5 \cdot \phi_1\right) \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

      4. *-commutative 44.8%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \cos \left(0.5 \cdot \phi_2\right) - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

      5. *-commutative 44.8%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \color{blue}{\left(\phi_2 \cdot 0.5\right)} - \sin \left(0.5 \cdot \phi_1\right) \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

      6. *-commutative 44.8%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \color{blue}{\left(\phi_1 \cdot 0.5\right)} \cdot \sin \left(0.5 \cdot \phi_2\right)\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

      7. *-commutative 44.8%

         \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \color{blue}{\left(\phi_2 \cdot 0.5\right)}\right), \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   10. Applied egg-rr 44.8%

       \[\leadsto {\left(\sqrt{\mathsf{hypot}\left(\left(\lambda_1 - \lambda_2\right) \cdot \color{blue}{\left(\cos \left(\phi_1 \cdot 0.5\right) \cdot \cos \left(\phi_2 \cdot 0.5\right) - \sin \left(\phi_1 \cdot 0.5\right) \cdot \sin \left(\phi_2 \cdot 0.5\right)\right)}, \phi_1 - \phi_2\right)} \cdot \sqrt{R}\right)}^{2} \]

   11. Taylor expanded in phi2 around inf 30.6%

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

       1. mul-1-neg 30.6%

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

       2. unsub-neg 30.6%

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

   13. Simplified 30.6%

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

   if 2.0500000000000001e165 < lambda2

   1. Initial program 41.6%

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

      1. hypot-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in phi2 around 0 81.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 81.3%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in lambda2 around inf 46.6%

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

   9. Taylor expanded in phi1 around 0 60.9%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 2.05 \cdot 10^{+165}:\\ \;\;\;\;\phi_2 \cdot \left(R - \frac{\phi_1 \cdot R}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 59.2% accurate, 23.5× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.25 \cdot 10^{+165}:\\ \;\;\;\;\phi_2 \cdot \left(R\_m - \phi_1 \cdot \frac{R\_m}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= lambda2 1.25e+165)
    (* phi2 (- R_m (* phi1 (/ R_m phi2))))
    (* lambda2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.25e+165) {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 1.25d+165) then
        tmp = phi2 * (r_m - (phi1 * (r_m / phi2)))
    else
        tmp = lambda2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 1.25e+165) {
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 1.25e+165:
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)))
	else:
		tmp = lambda2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 1.25e+165)
		tmp = Float64(phi2 * Float64(R_m - Float64(phi1 * Float64(R_m / phi2))));
	else
		tmp = Float64(lambda2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 1.25e+165)
		tmp = phi2 * (R_m - (phi1 * (R_m / phi2)));
	else
		tmp = lambda2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 1.25e+165], N[(phi2 * N[(R$95$m - N[(phi1 * N[(R$95$m / phi2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(lambda2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 1.24999999999999993e165

   1. Initial program 65.7%

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

      1. hypot-define 97.1%

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

   3. Simplified 97.1%

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

   5. Taylor expanded in phi2 around inf 30.6%

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

      1. mul-1-neg 30.6%

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

      2. unsub-neg 30.6%

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

      3. *-commutative 30.6%

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

      4. associate-/l* 30.6%

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

   7. Simplified 30.6%

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

   if 1.24999999999999993e165 < lambda2

   1. Initial program 41.6%

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

      1. hypot-define 93.6%

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

   3. Simplified 93.6%

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

   5. Taylor expanded in phi2 around 0 81.3%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 81.3%

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

   7. Simplified 81.3%

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

   8. Taylor expanded in lambda2 around inf 46.6%

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

   9. Taylor expanded in phi1 around 0 60.9%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 33.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 1.25 \cdot 10^{+165}:\\ \;\;\;\;\phi_2 \cdot \left(R - \phi_1 \cdot \frac{R}{\phi_2}\right)\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 49.2% accurate, 25.2× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.95 \cdot 10^{-175}:\\ \;\;\;\;\phi_1 \cdot \left(-R\_m\right)\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{+43}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (*
  R_s
  (if (<= phi2 1.95e-175)
    (* phi1 (- R_m))
    (if (<= phi2 3.3e+43) (* lambda2 R_m) (* phi2 R_m)))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.95e-175) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 3.3e+43) {
		tmp = lambda2 * R_m;
	} else {
		tmp = phi2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (phi2 <= 1.95d-175) then
        tmp = phi1 * -r_m
    else if (phi2 <= 3.3d+43) then
        tmp = lambda2 * r_m
    else
        tmp = phi2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (phi2 <= 1.95e-175) {
		tmp = phi1 * -R_m;
	} else if (phi2 <= 3.3e+43) {
		tmp = lambda2 * R_m;
	} else {
		tmp = phi2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if phi2 <= 1.95e-175:
		tmp = phi1 * -R_m
	elif phi2 <= 3.3e+43:
		tmp = lambda2 * R_m
	else:
		tmp = phi2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (phi2 <= 1.95e-175)
		tmp = Float64(phi1 * Float64(-R_m));
	elseif (phi2 <= 3.3e+43)
		tmp = Float64(lambda2 * R_m);
	else
		tmp = Float64(phi2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (phi2 <= 1.95e-175)
		tmp = phi1 * -R_m;
	elseif (phi2 <= 3.3e+43)
		tmp = lambda2 * R_m;
	else
		tmp = phi2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[phi2, 1.95e-175], N[(phi1 * (-R$95$m)), $MachinePrecision], If[LessEqual[phi2, 3.3e+43], N[(lambda2 * R$95$m), $MachinePrecision], N[(phi2 * R$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if phi2 < 1.94999999999999999e-175

   1. Initial program 65.4%

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

      1. hypot-define 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in phi1 around -inf 20.6%

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

      1. mul-1-neg 20.6%

         \[\leadsto \color{blue}{-R \cdot \phi_1} \]

      2. distribute-rgt-neg-in 20.6%

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

   7. Simplified 20.6%

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

   if 1.94999999999999999e-175 < phi2 < 3.3000000000000001e43

   1. Initial program 64.4%

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

      1. hypot-define 97.4%

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

   3. Simplified 97.4%

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

   5. Taylor expanded in phi2 around 0 91.6%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 91.6%

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

   7. Simplified 91.6%

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

   8. Taylor expanded in lambda2 around inf 21.1%

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

   9. Taylor expanded in phi1 around 0 24.7%

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

   if 3.3000000000000001e43 < phi2

   1. Initial program 55.8%

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

      1. hypot-define 92.6%

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

   3. Simplified 92.6%

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

   5. Taylor expanded in phi2 around inf 64.3%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\phi_2 \leq 1.95 \cdot 10^{-175}:\\ \;\;\;\;\phi_1 \cdot \left(-R\right)\\ \mathbf{elif}\;\phi_2 \leq 3.3 \cdot 10^{+43}:\\ \;\;\;\;\lambda_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\phi_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 36.6% accurate, 41.0× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3800000000000:\\ \;\;\;\;\phi_2 \cdot R\_m\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\_m\\ \end{array} \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (if (<= lambda2 3800000000000.0) (* phi2 R_m) (* lambda2 R_m))))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3800000000000.0) {
		tmp = phi2 * R_m;
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    real(8) :: tmp
    if (lambda2 <= 3800000000000.0d0) then
        tmp = phi2 * r_m
    else
        tmp = lambda2 * r_m
    end if
    code = r_s * tmp
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	double tmp;
	if (lambda2 <= 3800000000000.0) {
		tmp = phi2 * R_m;
	} else {
		tmp = lambda2 * R_m;
	}
	return R_s * tmp;
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	tmp = 0
	if lambda2 <= 3800000000000.0:
		tmp = phi2 * R_m
	else:
		tmp = lambda2 * R_m
	return R_s * tmp

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0
	if (lambda2 <= 3800000000000.0)
		tmp = Float64(phi2 * R_m);
	else
		tmp = Float64(lambda2 * R_m);
	end
	return Float64(R_s * tmp)
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp_2 = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = 0.0;
	if (lambda2 <= 3800000000000.0)
		tmp = phi2 * R_m;
	else
		tmp = lambda2 * R_m;
	end
	tmp_2 = R_s * tmp;
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * If[LessEqual[lambda2, 3800000000000.0], N[(phi2 * R$95$m), $MachinePrecision], N[(lambda2 * R$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if lambda2 < 3.8e12

   1. Initial program 64.3%

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

      1. hypot-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in phi2 around inf 19.7%

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

   if 3.8e12 < lambda2

   1. Initial program 60.3%

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

      1. hypot-define 95.3%

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

   3. Simplified 95.3%

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

   5. Taylor expanded in phi2 around 0 87.2%

      \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
   6. Step-by-step derivation

      1. *-commutative 87.2%

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

   7. Simplified 87.2%

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

   8. Taylor expanded in lambda2 around inf 38.6%

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

   9. Taylor expanded in phi1 around 0 46.6%

      \[\leadsto \color{blue}{R \cdot \lambda_2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\lambda_2 \leq 3800000000000:\\ \;\;\;\;\phi_2 \cdot R\\ \mathbf{else}:\\ \;\;\;\;\lambda_2 \cdot R\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 13.8% accurate, 109.7× speedup

Math

\[\begin{array}{l} R\_m = \left|R\right| \\ R\_s = \mathsf{copysign}\left(1, R\right) \\ [R_m, lambda1, lambda2, phi1, phi2] = \mathsf{sort}([R_m, lambda1, lambda2, phi1, phi2])\\ \\ R\_s \cdot \left(\lambda_2 \cdot R\_m\right) \end{array} \]

FPCore

R\_m = (fabs.f64 R)
R\_s = (copysign.f64 #s(literal 1 binary64) R)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
(FPCore (R_s R_m lambda1 lambda2 phi1 phi2)
 :precision binary64
 (* R_s (* lambda2 R_m)))

C

R\_m = fabs(R);
R\_s = copysign(1.0, R);
assert(R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2);
double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda2 * R_m);
}

Fortran

R\_m = abs(r)
R\_s = copysign(1.0d0, r)
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
real(8) function code(r_s, r_m, lambda1, lambda2, phi1, phi2)
    real(8), intent (in) :: r_s
    real(8), intent (in) :: r_m
    real(8), intent (in) :: lambda1
    real(8), intent (in) :: lambda2
    real(8), intent (in) :: phi1
    real(8), intent (in) :: phi2
    code = r_s * (lambda2 * r_m)
end function

Java

R\_m = Math.abs(R);
R\_s = Math.copySign(1.0, R);
assert R_m < lambda1 && lambda1 < lambda2 && lambda2 < phi1 && phi1 < phi2;
public static double code(double R_s, double R_m, double lambda1, double lambda2, double phi1, double phi2) {
	return R_s * (lambda2 * R_m);
}

Python

R\_m = math.fabs(R)
R\_s = math.copysign(1.0, R)
[R_m, lambda1, lambda2, phi1, phi2] = sort([R_m, lambda1, lambda2, phi1, phi2])
def code(R_s, R_m, lambda1, lambda2, phi1, phi2):
	return R_s * (lambda2 * R_m)

Julia

R\_m = abs(R)
R\_s = copysign(1.0, R)
R_m, lambda1, lambda2, phi1, phi2 = sort([R_m, lambda1, lambda2, phi1, phi2])
function code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	return Float64(R_s * Float64(lambda2 * R_m))
end

MATLAB

R\_m = abs(R);
R\_s = sign(R) * abs(1.0);
R_m, lambda1, lambda2, phi1, phi2 = num2cell(sort([R_m, lambda1, lambda2, phi1, phi2])){:}
function tmp = code(R_s, R_m, lambda1, lambda2, phi1, phi2)
	tmp = R_s * (lambda2 * R_m);
end

Wolfram

R\_m = N[Abs[R], $MachinePrecision]
R\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[R]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: R_m, lambda1, lambda2, phi1, and phi2 should be sorted in increasing order before calling this function.
code[R$95$s_, R$95$m_, lambda1_, lambda2_, phi1_, phi2_] := N[(R$95$s * N[(lambda2 * R$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.2%

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

   1. hypot-define 96.7%

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

3. Simplified 96.7%

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

5. Taylor expanded in phi2 around 0 89.9%

   \[\leadsto R \cdot \mathsf{hypot}\left(\color{blue}{\cos \left(0.5 \cdot \phi_1\right) \cdot \left(\lambda_1 - \lambda_2\right)}, \phi_1 - \phi_2\right) \]
6. Step-by-step derivation

   1. *-commutative 89.9%

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

7. Simplified 89.9%

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

8. Taylor expanded in lambda2 around inf 15.4%

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

9. Taylor expanded in phi1 around 0 15.0%

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

10. Final simplification 15.0%

    \[\leadsto \lambda_2 \cdot R \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024137 
(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))))))